Experimental Design, Data, and Data Summary

Size: px
Start display at page:

Download "Experimental Design, Data, and Data Summary"

Transcription

1 Chapter Six Experimental Design, Data, and Data Summary Tests of Hypotheses Because science advances by tests of hypotheses, scientists spend much of their time devising ways to test hypotheses. There are two general kinds of hypothesis tests; observational tests and experimental tests. Observational tests are those in which two or more groups observed to differ naturally in some respect are compared. For example, If you wished to test the hypothesis that bees are more effective pollinators than ants, you might compare the numbers of seeds set by flowers that bees chose to visit with the seed set of flowers that ants chose to visit. You would be comparing seed set for two groups of flowers that differ naturally in the identity of the organism that pollinated them. Similarly, to test the hypothesis that pollinators prefer blue flowers in a species in which flower color is variable, you could compare the number of visits received by groups of flowers that differ naturally in their color. A second way to test a hypothesis is to construct an experimental test. In an experimental test, the experimenter causes two otherwise similar groups to differ in order to test predictions about the factor that now differs. For example, you might wish to test the hypothesis that pollinators are more likely to visit flowers that are higher from the ground. An appropriate experimental test would be to grow a set of similar plants in pots and when they are in flower, set them out in a field with half on the ground and half perched on cement blocks. You would then observe your two sets of plants (low and high) and determine if the tall plants receive more visits. A question you might ask at this point is "Why bother to grow the plants in pots and place them at different heights - wouldn't it be easier to simply observe rates of pollinator visitation to naturally occurring plants of different height?". In other words, why not perform an observational test rather than an experimental test? In general, experimental tests provide more conclusive tests of hypotheses. Consider the following possibility; plants growing in wetter spots in a field may be taller and, many pollinating insects are susceptible to desiccation and therefore avoid areas of low humidity. Under these circumstances, taller plants may receive more visits but not because they are tall, but because they are in areas of higher humidity. The major weakness of observational tests of hypotheses is that groups being compared may differ in more than one aspect. From this, it is clear why an experimenter must only manipulate one factor at a time. In this way groups that differ in only one respect are compared. If, for example, one wished to test the hypothesis that both height and flower color affect pollinator visitation rates, one must perform two experiments; one in which only height differs between groups and one in which all flowers are presented at the same height and only flower color differs. Clearly, if the number pollinator visits to tall white flowers was compared to the number of visits to short blue flowers we would not know whether the difference was due to flower color only, to height only, or to both.

2 Elements of experimental design The basic elements of an experiment include treatments, replication, and randomization. Manipulations performed in the course of an experiment are referred to as treatments, and the manipulated groups of subjects are called treatment groups. Although only one factor will differ among treatment groups, there may be more than two groups (e.g. tall, medium tall, medium short, and short). Some experiments are designed to compare a manipulated group with an unmanipulated group called a control. For example, one might hypothesize that bees are attracted to the smell of lemons. An experimental test of this hypothesis might involve a comparison of the frequency of bee visits to a set of flowers, half of which have been sprayed with lemon juice. A second set of flowers sprayed with plain water would constitute the control group. They must be identical to the treatment group in all respects except the treatment factor. Other experiments may be designed to compare two treatment groups. For example, suppose you wanted to test the hypothesis that bees are more attracted to the scent of lemon than the scent of beer. In this case, you might design an experiment with two treatment groups, one consisting of flowers sprayed with lemon and the other of flowers sprayed with beer. Treatments must be applied to more than one experimental unit. This is because those units are likely to differ naturally in ways that influence your results. Consider the experiment to determine if bees are attracted to lemon. If you used only one lemonsprayed flower and one control flower, bees might visit these flowers at different because the two flowers happen to differ in age or size. You can have much greater confidence that your treatments are truly responsible for the difference if you use several flowers. The multiple experimental units to which treatments are applied are called replicates. The ideal number of replicates for an experiment will depend on how much natural variation there is among experimental units and how sensitive you want your experiment to be. In order to avoid confounding your treatments with natural variation, replicates should be assigned to treatments randomly. Imagine that you go out into the field to conduct your experiment to determine if bees are attracted to lemon scent. You designate the first 20 flowers you encounter to be treatment flowers and the next 20 to be controls. Now imagine that you came to your field site from the west in the afternoon. The first flowers you encounter will probably be in the sun and the flowers you encounter later might be in the shade. Bees are known to prefer to forage in the sun, so this non-random assignment of replicates to treatments will influence your results. A better procedure would be to flip a coin as you encounter each flower and designate the flower as a treatment flower if the coin comes up heads and as a control flower if it comes up tails. The assignment of replicates to treatments at random is called randomization. Data All science is based on data of some kind (note that the word "data" is the plural of "datum"). The data may come from the results of experiments or from noting patterns in nature or it may even be purely theoretical, as in Einstein and Bohr's famous letters on thought experiments in physics. Data are generally obtained as individual observations in which a particular aspect or variable is quantified. Different observations of the same

3 variable can constitute replicates. A sample consists of a collection of replicate observations (data) selected by a specified procedure from a larger population. Before we go any further, we need to distinguish between different kinds of data and discuss how you describe the samples you have collected and how you use samples of data to test specific hypotheses. Kinds of Data Data can be categorized in different ways, and different types of data are often analyzed using different statistical techniques. Nominal data fall into distinct categories that cannot be ordered numerically. For example, the pollinators visiting a plant might fall into three categories: bees, butterflies, and wasps. Or flowers of a plant species might be red, blue, or white. These are categories that can be named (hence the designation nominal ), but the categories do not bear any numerical relationship to one another. Ordinal data are data that can be ordered numerically. For example, plants could be divided into those bearing 4, 5, or 6 flowers, categories that can clearly be ordered from smallest to largest. Or one could measure the heights of a number of plants and obtain ordinal data (e.g cm, 5.7 cm, 12.4 cm). Ordinal data can be further divided into discrete and continuous data. Discrete data fall into categories that have distinct boundaries. In this sense, they resemble nominal data, which are always discrete, but unlike nominal data, discrete ordinal data can be ordered numerically. The flower number data above are discrete data because you do not find, for example, plants with 1.6 flowers or 4.3 flowers. In contrast, continuous data fall along a continuum like sizes or time measurements. For example, the four bees visiting a flower may be 8.2, 9.1, 9.2, and 10.3 mm in length and spend 2.3, 4.1, 6.0, and 3.2 seconds on the flower. The distinction between different kinds of data is not always clear cut. In some cases, one may have a choice of whether to treat data as continuous or discrete. Consider data on flowers, which may be categorized nominally as either blue or white. If there is variation in the intensity of blue, it might be possible to establish several discrete ordinal categories; white, light, medium. and dark blue. With the appropriate equipment, you could quantify color on a continuous scale by the amount of blue light reflected by each flower. Your choice of what kind of data to collect would depend on the questions you want to answer. The difference between discrete and continuous ordinal data can be particularly subtle (to the point of invisibility), but an easy way of thinking about it is whether or not the data could potentially be divided further by use of better measuring tools. If your subjects truly form two groups, blue and white, then no matter how you measure individuals, they still come out either one or the other. However, in the more realistic case in which individuals vary more or less continuously from blue to white, the number of categories is limited only by the precision with which you measure the color. Different types of data may be summarized and analyzed in different ways. Most of the remaining discussion in this chapter will deal with ordinal continuous data. We will return to ordinal discrete and nominal data when we discuss the Chi-square procedure (Chapter 7). Describing Data Sets: Means One of the first things to do with a sample of data is to describe it in a simple fashion that will both give others some idea of its nature without their having to see the data

4 themselves and allow it to be compared to other samples. For continuous data, the two simplest descriptors are some value indicating what an "average" value is (e.g. mean, median, or mode) and some measure of the amount of variation around that average. The most commonly used "average" value is the mean, the sum of all the data values (which we call "x" s) divided by the number of values (symbolized "n"). mean = X - = Sx n Before we go further, let's get a set of data to use as an example. Consider a population of bean plants. A scientist has noticed that the flowers are visited by both butterflies and bees (the observation). She knows that bees collect lots of pollen and so wonders whether they are better at pollinating the beans than the butterflies (the question), and she guesses that they do (an informal hypothesis). Formally stated, a single bee visiting a flower will pollinate with a higher efficiency than a single butterfly, thus producing a greater number of seeds in the bean pod (a hypothesis). As a test of the hypothesis, she removes all the open flowers on several plants, then covers the plants with pollinator-exclusion cages for three days. She then returns, removes the cages and watches pollinators visit the flowers. After a single butterfly or bee visits a flower, she bags that flower with a net bag to prevent further visitation. After several hours, she decides that she has sufficient data and goes home. Two weeks later she comes back and collects the developing pods from her marked flowers. She carefully dissects the pods and counts the number of developing seeds in each. In this fashion she obtains the following data: Number of seeds per pod Bee pollinated Butterfly pollinated mean (X - ) sample size (n) First of all, note another descriptor of the two samples: the sample size. This information is of interest because the more replicates we have, the greater our confidence that any differences we observe between treatment groups is really due to treatments. From the sample sizes here, we can see that more bees were active, resulting in 14 observations, as opposed to 12 for the butterfly-pollinated flowers.

5 Inspection of the means gives the simplest information about the data and suggests that butterflies are actually better pollinators than bees. But always remember that this difference might be just due to chance events. In Chapter seven we will discuss how to determine whether these means are statistically significantly different using a t-test. Describing Data Sets: Dispersion about the Mean Another useful description of the data is how much the data are spread out around the mean. Two sets of data may have the same mean and yet be quite different. In one set, every data point may have the mean value ( 4,4,4,4,4,4,4). A second set may have the same mean, but each data point is different from every other (1,2,3,4,5,6,7). One way to describe the difference in how data are dispersed around the mean is to report the range: the minimum and maximum values in the sample. However, the most frequently used measurement for describing the dispersion of data is the variance, symbolized s 2. (The variance of a sample of data is equal to the square of the standard deviation, another commonly used measure of variation). Higher values for the variance indicate that the data are dispersed widely around the mean; lower values indicate that the values are all close to the mean. The variance is calculated as: s 2 = S x 2 - (Sx)2 n n-1 Frequency Distributions Graphs are very valuable tools for summarizing data. Scientists, like everybody else, find it difficult to pick out complex patterns in data just by looking at the numbers. They often have to sit back, graph out the data, and just stare at it for a while. A frequency histogram is a graph that illustrates the distribution of data around the mean. Traditionally, the vertical axis (the y-axis) gives the frequency of occurrence of a particular measurement and the horizontal axis (the x-axis) represents your variable of interest (e.g., number of developing seeds). Figure 6.1 shows the frequency histograms for the data from experiment on the efficiency of bees and bumblebees described above. The frequency histogram in Figure 6.1 shows that the two samples not only have different means but that they also have very different distributions around the mean. The distribution for bees is symmetric about the mean and is called a normal distribution (symmetry is only one of several criteria for a normal distribution, see below). It is a version of the bell-shaped curve that you may have heard instructors mention in the past with respect to grades. In the butterfly distribution, most of the values are packed into the right-hand portion of the graph, and the data are spread

6 6 5 BEE POLLINATED NUMBER OF FLOWERS BUTTERFLY POLLINATED NUMBER OF FLOWERS NUMBER OF SEEDS/FLOWER Figure 6.1. Frequency histograms for bean pollination data. over a slightly larger range of seeds/pod. This sort of distribution is "skewed" and, confusingly enough, in this case is left or negative-skewed (the direction of skewness is determined by the location of the tail of the distribution). Other types of distributions are also possible. For example, a uniform distribution would have an equal number of observations for each unit of measurement. For example, the sample might have been 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, and 6 (trying plotting this distribution, either on paper or in your head). A frequency histogram in which the data form two peaks is said to show a bimodal distribution. The data you collect are likely to be normal or somewhat skewed but will always be spread out on either side of a "central" point (the mean). When you analyze your data, you are often interested in this central point and that is why we calculate means. However, the mean is not necessarily truly the central point. In fact, the mean, reported alone, may not be very meaningful. If your plotted data were skewed or bimodal, the

7 mean might be nowhere near the center of the range of the data. For example, the most common measurement for the butterfly data was eight, yet the mean is much less (6.50). It is important to note here that many statistical tests require that your data be normally distributed. The further your data deviate from normality, the less reliable the result of the test. Many statistical procedures can be used either to test for normality in your data or to transform your data to make them normally distributed (for example, taking the logarithm of each datum). Or, if your data are severely nonnormal, another whole class of statistics ("nonparametric" statistics) can be used. However, we are not going to study those tests here: if more sophisticated statistics are necessary for your analysis, your Teaching Assistant will help you. The Metric System A final note. The United States is virtually the last country on earth to switch over seriously to the metric system. The scientific community, on the other hand, is much less affected by national boundaries and switched nearly completely years ago. No major scientific publication will consider research using anything but the metric system, and neither will we. All your units must be given in grams, meters, and liters, or decimal fractions thereof, with the correct abbreviations where necessary. We don't care about the units you use in the field. I have known scientists who forgot to bring equipment to the field and were forced to take data in "beer-bottle units," but, before the data are analyzed and presented in your paper, they must be converted to metric units.

Probability Distributions

Probability Distributions Probability Distributions Probability This is not a math class, or an applied math class, or a statistics class; but it is a computer science course! Still, probability, which is a math-y concept underlies

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT

More information

Background to Statistics

Background to Statistics FACT SHEET Background to Statistics Introduction Statistics include a broad range of methods for manipulating, presenting and interpreting data. Professional scientists of all kinds need to be proficient

More information

Do students sleep the recommended 8 hours a night on average?

Do students sleep the recommended 8 hours a night on average? BIEB100. Professor Rifkin. Notes on Section 2.2, lecture of 27 January 2014. Do students sleep the recommended 8 hours a night on average? We first set up our null and alternative hypotheses: H0: μ= 8

More information

Scientific Inquiry. Deborah Turner

Scientific Inquiry. Deborah Turner Scientific Inquiry Deborah Turner CHAPTER 1 Scientific Inquiry The Scientific inquiry is a process of steps that scientist use in order to explain and idea. Steps normally are in order for the following

More information

9/2/2010. Wildlife Management is a very quantitative field of study. throughout this course and throughout your career.

9/2/2010. Wildlife Management is a very quantitative field of study. throughout this course and throughout your career. Introduction to Data and Analysis Wildlife Management is a very quantitative field of study Results from studies will be used throughout this course and throughout your career. Sampling design influences

More information

Honey, It's Electric: Bees Sense Charge On Flowers

Honey, It's Electric: Bees Sense Charge On Flowers Honey, It's Electric: Bees Sense Charge On Flowers Flowers are nature's ad men. They'll do anything to attract the attention of the pollinators that help them reproduce. That means spending precious energy

More information

Pollination Lab Bio 220 Ecology and Evolution Fall, 2016

Pollination Lab Bio 220 Ecology and Evolution Fall, 2016 Pollination Lab Bio 220 Ecology and Evolution Fall, 2016 Journal reading: Comparison of pollen transfer dynamics by multiple floral visitors: experiments with pollen and fluorescent dye Introduction: Flowers

More information

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur

Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Probability Methods in Civil Engineering Prof. Dr. Rajib Maity Department of Civil Engineering Indian Institution of Technology, Kharagpur Lecture No. # 36 Sampling Distribution and Parameter Estimation

More information

SESSION 5 Descriptive Statistics

SESSION 5 Descriptive Statistics SESSION 5 Descriptive Statistics Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample and the measures. Together with simple

More information

MATH 1150 Chapter 2 Notation and Terminology

MATH 1150 Chapter 2 Notation and Terminology MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the

More information

1-1. Chapter 1. Sampling and Descriptive Statistics by The McGraw-Hill Companies, Inc. All rights reserved.

1-1. Chapter 1. Sampling and Descriptive Statistics by The McGraw-Hill Companies, Inc. All rights reserved. 1-1 Chapter 1 Sampling and Descriptive Statistics 1-2 Why Statistics? Deal with uncertainty in repeated scientific measurements Draw conclusions from data Design valid experiments and draw reliable conclusions

More information

Statistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018

Statistics Boot Camp. Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 Statistics Boot Camp Dr. Stephanie Lane Institute for Defense Analyses DATAWorks 2018 March 21, 2018 Outline of boot camp Summarizing and simplifying data Point and interval estimation Foundations of statistical

More information

Data set B is 2, 3, 3, 3, 5, 8, 9, 9, 9, 15. a) Determine the mean of the data sets. b) Determine the median of the data sets.

Data set B is 2, 3, 3, 3, 5, 8, 9, 9, 9, 15. a) Determine the mean of the data sets. b) Determine the median of the data sets. FOUNDATIONS OF MATH 11 Ch. 5 Day 1: EXPLORING DATA VOCABULARY A measure of central tendency is a value that is representative of a set of numerical data. These values tend to lie near the middle of a set

More information

Pea Patch Pollination Game

Pea Patch Pollination Game Pea Patch Pollination Game Classroom Activity: 5-8 Time: One 45-60-minute class period Overview: In this activity, students play a simulation game modeling changes in a plant population (a Pea Patch) caused

More information

AP Final Review II Exploring Data (20% 30%)

AP Final Review II Exploring Data (20% 30%) AP Final Review II Exploring Data (20% 30%) Quantitative vs Categorical Variables Quantitative variables are numerical values for which arithmetic operations such as means make sense. It is usually a measure

More information

Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras

Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Lecture - 21 Central Potential and Central Force Ready now to take up the idea

More information

Visual Displays of Information in Understanding Evolution by Natural Selection

Visual Displays of Information in Understanding Evolution by Natural Selection Name: Date: Visual Displays of Information in Understanding Evolution by Natural Selection The alpine skypilot is a purple perennial wildflower that is native to western North America. It grows in dry

More information

1 Normal Distribution.

1 Normal Distribution. Normal Distribution.. Introduction A Bernoulli trial is simple random experiment that ends in success or failure. A Bernoulli trial can be used to make a new random experiment by repeating the Bernoulli

More information

Listening. The Air. Did you know? Did you know?

Listening. The Air. Did you know? Did you know? Listening 1. Find a place to sit. 2. Close your eyes and listen carefully to all the sounds you can hear. Cup your hands around your ears and turn your head to help you listen in particular directions.

More information

THE SAMPLING DISTRIBUTION OF THE MEAN

THE SAMPLING DISTRIBUTION OF THE MEAN THE SAMPLING DISTRIBUTION OF THE MEAN COGS 14B JANUARY 26, 2017 TODAY Sampling Distributions Sampling Distribution of the Mean Central Limit Theorem INFERENTIAL STATISTICS Inferential statistics: allows

More information

Inferential statistics

Inferential statistics Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample. Ahmed-Refat-ZU Null and alternative hypotheses In hypotheses testing,

More information

Module 03 Lecture 14 Inferential Statistics ANOVA and TOI

Module 03 Lecture 14 Inferential Statistics ANOVA and TOI Introduction of Data Analytics Prof. Nandan Sudarsanam and Prof. B Ravindran Department of Management Studies and Department of Computer Science and Engineering Indian Institute of Technology, Madras Module

More information

Unit 1: Introduction to Chemistry

Unit 1: Introduction to Chemistry Unit 1: Introduction to Chemistry I. Observations vs. Inferences Observation: information you gather using your five senses ***You will NEVER use taste in class! o Describes facts Examples You see the

More information

Lesson Adaptation Activity: Developing and Using Models

Lesson Adaptation Activity: Developing and Using Models Lesson Adaptation Activity: Developing and Using Models Related MA STE Framework Standard: 2-LS2-2. Develop a simple model that mimics the function of an animal in dispersing seeds or pollinating plants.*

More information

Types of Information. Topic 2 - Descriptive Statistics. Examples. Sample and Sample Size. Background Reading. Variables classified as STAT 511

Types of Information. Topic 2 - Descriptive Statistics. Examples. Sample and Sample Size. Background Reading. Variables classified as STAT 511 Topic 2 - Descriptive Statistics STAT 511 Professor Bruce Craig Types of Information Variables classified as Categorical (qualitative) - variable classifies individual into one of several groups or categories

More information

Part 3: Parametric Models

Part 3: Parametric Models Part 3: Parametric Models Matthew Sperrin and Juhyun Park August 19, 2008 1 Introduction There are three main objectives to this section: 1. To introduce the concepts of probability and random variables.

More information

Food Chains. energy: what is needed to do work or cause change

Food Chains. energy: what is needed to do work or cause change Have you ever seen a picture that shows a little fish about to be eaten by a big fish? Sometimes the big fish has an even bigger fish behind it. This is a simple food chain. A food chain is the path of

More information

Review of Statistics 101

Review of Statistics 101 Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods

More information

2nd Grade. Slide 1 / 106. Slide 2 / 106. Slide 3 / 106. Plants. Table of Contents

2nd Grade. Slide 1 / 106. Slide 2 / 106. Slide 3 / 106. Plants. Table of Contents Slide 1 / 106 Slide 2 / 106 2nd Grade Plants 2015-11-24 www.njctl.org Table of Contents Slide 3 / 106 Click on the topic to go to that section What are plants? Photosynthesis Pollination Dispersal Slide

More information

Discrete Probability. Chemistry & Physics. Medicine

Discrete Probability. Chemistry & Physics. Medicine Discrete Probability The existence of gambling for many centuries is evidence of long-running interest in probability. But a good understanding of probability transcends mere gambling. The mathematics

More information

3. DISCRETE PROBABILITY DISTRIBUTIONS

3. DISCRETE PROBABILITY DISTRIBUTIONS 1 3. DISCRETE PROBABILITY DISTRIBUTIONS Probability distributions may be discrete or continuous. This week we examine two discrete distributions commonly used in biology: the binomial and Poisson distributions.

More information

Special Theory of Relativity Prof. Shiva Prasad Department of Physics Indian Institute of Technology, Bombay. Lecture - 15 Momentum Energy Four Vector

Special Theory of Relativity Prof. Shiva Prasad Department of Physics Indian Institute of Technology, Bombay. Lecture - 15 Momentum Energy Four Vector Special Theory of Relativity Prof. Shiva Prasad Department of Physics Indian Institute of Technology, Bombay Lecture - 15 Momentum Energy Four Vector We had started discussing the concept of four vectors.

More information

2nd Grade. Plants.

2nd Grade. Plants. 1 2nd Grade Plants 2015 11 24 www.njctl.org 2 Table of Contents Click on the topic to go to that section What are plants? Photosynthesis Pollination Dispersal 3 Lab: What do plants need? What do plants

More information

Elementary Statistics

Elementary Statistics Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q:

More information

Part III: Unstructured Data. Lecture timetable. Analysis of data. Data Retrieval: III.1 Unstructured data and data retrieval

Part III: Unstructured Data. Lecture timetable. Analysis of data. Data Retrieval: III.1 Unstructured data and data retrieval Inf1-DA 2010 20 III: 28 / 89 Part III Unstructured Data Data Retrieval: III.1 Unstructured data and data retrieval Statistical Analysis of Data: III.2 Data scales and summary statistics III.3 Hypothesis

More information

This is particularly true if you see long tails in your data. What are you testing? That the two distributions are the same!

This is particularly true if you see long tails in your data. What are you testing? That the two distributions are the same! Two sample tests (part II): What to do if your data are not distributed normally: Option 1: if your sample size is large enough, don't worry - go ahead and use a t-test (the CLT will take care of non-normal

More information

Community Involvement in Research Monitoring Pollinator Populations using Public Participation in Scientific Research

Community Involvement in Research Monitoring Pollinator Populations using Public Participation in Scientific Research Overview Community Involvement in Research Monitoring Pollinator Populations using Public Participation in Scientific Research Public Participation in Scientific Research (PPSR) is a concept adopted by

More information

Last Lecture. Distinguish Populations from Samples. Knowing different Sampling Techniques. Distinguish Parameters from Statistics

Last Lecture. Distinguish Populations from Samples. Knowing different Sampling Techniques. Distinguish Parameters from Statistics Last Lecture Distinguish Populations from Samples Importance of identifying a population and well chosen sample Knowing different Sampling Techniques Distinguish Parameters from Statistics Knowing different

More information

Please bring the task to your first physics lesson and hand it to the teacher.

Please bring the task to your first physics lesson and hand it to the teacher. Pre-enrolment task for 2014 entry Physics Why do I need to complete a pre-enrolment task? This bridging pack serves a number of purposes. It gives you practice in some of the important skills you will

More information

B.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba

B.N.Bandodkar College of Science, Thane. Random-Number Generation. Mrs M.J.Gholba B.N.Bandodkar College of Science, Thane Random-Number Generation Mrs M.J.Gholba Properties of Random Numbers A sequence of random numbers, R, R,., must have two important statistical properties, uniformity

More information

Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D.

Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D. Mitosis Data Analysis: Testing Statistical Hypotheses By Dana Krempels, Ph.D. and Steven Green, Ph.D. The number of cells in various stages of mitosis in your treatment and control onions are your raw

More information

Confidence Intervals

Confidence Intervals Quantitative Foundations Project 3 Instructor: Linwei Wang Confidence Intervals Contents 1 Introduction 3 1.1 Warning....................................... 3 1.2 Goals of Statistics..................................

More information

Glossary for the Triola Statistics Series

Glossary for the Triola Statistics Series Glossary for the Triola Statistics Series Absolute deviation The measure of variation equal to the sum of the deviations of each value from the mean, divided by the number of values Acceptance sampling

More information

(Sessions I and II)* BROWARD COUNTY ELEMENTARY SCIENCE BENCHMARK PLAN FOR PERSONAL USE

(Sessions I and II)* BROWARD COUNTY ELEMENTARY SCIENCE BENCHMARK PLAN FOR PERSONAL USE activities 19&20 What Do Plants Need? (Sessions I and II)* BROWARD COUNTY ELEMENTARY SCIENCE BENCHMARK PLAN Grade 1 Quarter 2 Activities 19 & 20 SC.A.1.1.1 The student knows that objects can be described,

More information

CHAPTER 13: F PROBABILITY DISTRIBUTION

CHAPTER 13: F PROBABILITY DISTRIBUTION CHAPTER 13: F PROBABILITY DISTRIBUTION continuous probability distribution skewed to the right variable values on horizontal axis are 0 area under the curve represents probability horizontal asymptote

More information

Ø Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.

Ø Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization. Statistical Tools in Evaluation HPS 41 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific number

More information

Chapter 2: Tools for Exploring Univariate Data

Chapter 2: Tools for Exploring Univariate Data Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 2: Tools for Exploring Univariate Data Section 2.1: Introduction What is

More information

Violating the normal distribution assumption. So what do you do if the data are not normal and you still need to perform a test?

Violating the normal distribution assumption. So what do you do if the data are not normal and you still need to perform a test? Violating the normal distribution assumption So what do you do if the data are not normal and you still need to perform a test? Remember, if your n is reasonably large, don t bother doing anything. Your

More information

6 th Grade Unit 1 Benchmark: Test on Tuesday, November 10 th

6 th Grade Unit 1 Benchmark: Test on Tuesday, November 10 th 6 th Grade Unit 1 Benchmark: Test on Tuesday, November 10 th Engineering Design Process - If you think about it, the order of the steps makes sense, so think logically! The test will ask you to identify

More information

Lab Slide Rules and Log Scales

Lab Slide Rules and Log Scales Name: Lab Slide Rules and Log Scales [EER Note: This is a much-shortened version of my lab on this topic. You won t finish, but try to do one of each type of calculation if you can. I m available to help.]

More information

Lecture 1 : Basic Statistical Measures

Lecture 1 : Basic Statistical Measures Lecture 1 : Basic Statistical Measures Jonathan Marchini October 11, 2004 In this lecture we will learn about different types of data encountered in practice different ways of plotting data to explore

More information

Topic 1. Definitions

Topic 1. Definitions S Topic. Definitions. Scalar A scalar is a number. 2. Vector A vector is a column of numbers. 3. Linear combination A scalar times a vector plus a scalar times a vector, plus a scalar times a vector...

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx

More information

Pollinator Adaptations

Pollinator Adaptations Adapted from: Life Lab Garden Pollinators unit Pollinator Adaptations Overview: Students will learn about pollinators and their adaptations, and match flowers to the kinds of pollinators they attract.

More information

Sampling, Frequency Distributions, and Graphs (12.1)

Sampling, Frequency Distributions, and Graphs (12.1) 1 Sampling, Frequency Distributions, and Graphs (1.1) Design: Plan how to obtain the data. What are typical Statistical Methods? Collect the data, which is then subjected to statistical analysis, which

More information

ST Presenting & Summarising Data Descriptive Statistics. Frequency Distribution, Histogram & Bar Chart

ST Presenting & Summarising Data Descriptive Statistics. Frequency Distribution, Histogram & Bar Chart ST2001 2. Presenting & Summarising Data Descriptive Statistics Frequency Distribution, Histogram & Bar Chart Summary of Previous Lecture u A study often involves taking a sample from a population that

More information

ABE Math Review Package

ABE Math Review Package P a g e ABE Math Review Package This material is intended as a review of skills you once learned and wish to review before your assessment. Before studying Algebra, you should be familiar with all of the

More information

Descriptive Statistics (And a little bit on rounding and significant digits)

Descriptive Statistics (And a little bit on rounding and significant digits) Descriptive Statistics (And a little bit on rounding and significant digits) Now that we know what our data look like, we d like to be able to describe it numerically. In other words, how can we represent

More information

Chapter 26: Comparing Counts (Chi Square)

Chapter 26: Comparing Counts (Chi Square) Chapter 6: Comparing Counts (Chi Square) We ve seen that you can turn a qualitative variable into a quantitative one (by counting the number of successes and failures), but that s a compromise it forces

More information

Weather Observations. Weather Observations. 1 of 10. Copyright 2007, Exemplars, Inc. All rights reserved.

Weather Observations. Weather Observations. 1 of 10. Copyright 2007, Exemplars, Inc. All rights reserved. As we begin our unit on weather, we will go outside and observe as many things as we can about the weather today using our senses. On your recording sheet, describe what you observe in detail and use what

More information

Take the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8", how accurate is our result?

Take the measurement of a person's height as an example. Assuming that her height has been determined to be 5' 8, how accurate is our result? Error Analysis Introduction The knowledge we have of the physical world is obtained by doing experiments and making measurements. It is important to understand how to express such data and how to analyze

More information

Chapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67

Chapter 6. The Standard Deviation as a Ruler and the Normal Model 1 /67 Chapter 6 The Standard Deviation as a Ruler and the Normal Model 1 /67 Homework Read Chpt 6 Complete Reading Notes Do P129 1, 3, 5, 7, 15, 17, 23, 27, 29, 31, 37, 39, 43 2 /67 Objective Students calculate

More information

Chapter 4a Probability Models

Chapter 4a Probability Models Chapter 4a Probability Models 4a.2 Probability models for a variable with a finite number of values 297 4a.1 Introduction Chapters 2 and 3 are concerned with data description (descriptive statistics) where

More information

4/1/2012. Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20. Skipping Topics 11 and 15. Topic 12. Normal Distribution

4/1/2012. Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20. Skipping Topics 11 and 15. Topic 12. Normal Distribution Test 2 Covers Topics 12, 13, 16, 17, 18, 14, 19 and 20 Skipping Topics 11 and 15 Topic 12 Normal Distribution 1 Normal Distribution If Density Curve is symmetric, single peaked, bell-shaped then it is

More information

Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify

More information

ADMS2320.com. We Make Stats Easy. Chapter 4. ADMS2320.com Tutorials Past Tests. Tutorial Length 1 Hour 45 Minutes

ADMS2320.com. We Make Stats Easy. Chapter 4. ADMS2320.com Tutorials Past Tests. Tutorial Length 1 Hour 45 Minutes We Make Stats Easy. Chapter 4 Tutorial Length 1 Hour 45 Minutes Tutorials Past Tests Chapter 4 Page 1 Chapter 4 Note The following topics will be covered in this chapter: Measures of central location Measures

More information

- measures the center of our distribution. In the case of a sample, it s given by: y i. y = where n = sample size.

- measures the center of our distribution. In the case of a sample, it s given by: y i. y = where n = sample size. Descriptive Statistics: One of the most important things we can do is to describe our data. Some of this can be done graphically (you should be familiar with histograms, boxplots, scatter plots and so

More information

Chapter 3. Data Description

Chapter 3. Data Description Chapter 3. Data Description Graphical Methods Pie chart It is used to display the percentage of the total number of measurements falling into each of the categories of the variable by partition a circle.

More information

MA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.

MA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table. MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at

More information

Draft Proof - Do not copy, post, or distribute

Draft Proof - Do not copy, post, or distribute 1 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1. Distinguish between descriptive and inferential statistics. Introduction to Statistics 2. Explain how samples and populations,

More information

MATH 10 INTRODUCTORY STATISTICS

MATH 10 INTRODUCTORY STATISTICS MATH 10 INTRODUCTORY STATISTICS Tommy Khoo Your friendly neighbourhood graduate student. Week 1 Chapter 1 Introduction What is Statistics? Why do you need to know Statistics? Technical lingo and concepts:

More information

Statistics 301: Probability and Statistics Introduction to Statistics Module

Statistics 301: Probability and Statistics Introduction to Statistics Module Statistics 301: Probability and Statistics Introduction to Statistics Module 1 2018 Introduction to Statistics Statistics is a science, not a branch of mathematics, but uses mathematical models as essential

More information

ACTIVITY 3. Learning Targets: 38 Unit 1 Equations and Inequalities. Solving Inequalities. continued. My Notes

ACTIVITY 3. Learning Targets: 38 Unit 1 Equations and Inequalities. Solving Inequalities. continued. My Notes Learning Targets: Write inequalities to represent real-world situations. Solve multi-step inequalities. SUGGESTED LEARNING STRATEGIES: Create Representations, Guess and Check, Look for a Pattern, Think-Pair-Share,

More information

Algebra 1: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney

Algebra 1: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney Algebra 1: Semester 2 Practice Final Unofficial Worked Out Solutions by Earl Whitney 1. The situation described in this problem involves probability without replacement. Because of this, the probabilities

More information

Standards Alignment... 5 Safe Science... 9 Scienti c Inquiry...11 Assembling Rubber Band Books... 15

Standards Alignment... 5 Safe Science... 9 Scienti c Inquiry...11 Assembling Rubber Band Books... 15 Standards Alignment... 5 Safe Science... 9 Scienti c Inquiry...11 Assembling Rubber Band Books... 15 Organization and Development of Living Organisms Enviroscape... 17 Plant Parts...23 Getting to the Root

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

Science Process Skills Home Learning

Science Process Skills Home Learning Science Process Skills Home Learning Modified True/False Indicate whether the statement is true or false. If false, change the identified word or phrase to make the statement true. 1 An explanation backed

More information

P (E) = P (A 1 )P (A 2 )... P (A n ).

P (E) = P (A 1 )P (A 2 )... P (A n ). Lecture 9: Conditional probability II: breaking complex events into smaller events, methods to solve probability problems, Bayes rule, law of total probability, Bayes theorem Discrete Structures II (Summer

More information

Shape, Outliers, Center, Spread Frequency and Relative Histograms Related to other types of graphical displays

Shape, Outliers, Center, Spread Frequency and Relative Histograms Related to other types of graphical displays Histograms: Shape, Outliers, Center, Spread Frequency and Relative Histograms Related to other types of graphical displays Sep 9 1:13 PM Shape: Skewed left Bell shaped Symmetric Bi modal Symmetric Skewed

More information

MITOCW ocw-18_02-f07-lec02_220k

MITOCW ocw-18_02-f07-lec02_220k MITOCW ocw-18_02-f07-lec02_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.

More information

Insect Investigations

Insect Investigations Investigative Questions What are some adaptations that insects have that help them to feed on different foods and from different parts of plants, especially flowers? Goal: Students explore the ways that

More information

Chapter 4. Displaying and Summarizing. Quantitative Data

Chapter 4. Displaying and Summarizing. Quantitative Data STAT 141 Introduction to Statistics Chapter 4 Displaying and Summarizing Quantitative Data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 31 4.1 Histograms 1 We divide the range

More information

Dedicated to bees, and My dad who loved reading to the kids on his lap, My children who spent plenty of time on his lap and mine, And Melvin, who

Dedicated to bees, and My dad who loved reading to the kids on his lap, My children who spent plenty of time on his lap and mine, And Melvin, who Dedicated to bees, and My dad who loved reading to the kids on his lap, My children who spent plenty of time on his lap and mine, And Melvin, who thinks all laps belong to him. Published by Melvin TC PO

More information

Notes Week 2 Chapter 3 Probability WEEK 2 page 1

Notes Week 2 Chapter 3 Probability WEEK 2 page 1 Notes Week 2 Chapter 3 Probability WEEK 2 page 1 The sample space of an experiment, sometimes denoted S or in probability theory, is the set that consists of all possible elementary outcomes of that experiment

More information

1 Measurement Uncertainties

1 Measurement Uncertainties 1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise.

More information

Glossary. The ISI glossary of statistical terms provides definitions in a number of different languages:

Glossary. The ISI glossary of statistical terms provides definitions in a number of different languages: Glossary The ISI glossary of statistical terms provides definitions in a number of different languages: http://isi.cbs.nl/glossary/index.htm Adjusted r 2 Adjusted R squared measures the proportion of the

More information

2. Probability. Chris Piech and Mehran Sahami. Oct 2017

2. Probability. Chris Piech and Mehran Sahami. Oct 2017 2. Probability Chris Piech and Mehran Sahami Oct 2017 1 Introduction It is that time in the quarter (it is still week one) when we get to talk about probability. Again we are going to build up from first

More information

Chapter 1 Statistical Inference

Chapter 1 Statistical Inference Chapter 1 Statistical Inference causal inference To infer causality, you need a randomized experiment (or a huge observational study and lots of outside information). inference to populations Generalizations

More information

Harvard University. Rigorous Research in Engineering Education

Harvard University. Rigorous Research in Engineering Education Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected

More information

Sampling Populations limited in the scope enumerate

Sampling Populations limited in the scope enumerate Sampling Populations Typically, when we collect data, we are somewhat limited in the scope of what information we can reasonably collect Ideally, we would enumerate each and every member of a population

More information

Pre-Lab 0.2 Reading: Measurement

Pre-Lab 0.2 Reading: Measurement Name Block Pre-Lab 0.2 Reading: Measurement section 1 Description and Measurement Before You Read Weight, height, and length are common measurements. List at least five things you can measure. What You

More information

MIT BLOSSOMS INITIATIVE

MIT BLOSSOMS INITIATIVE MIT BLOSSOMS INITIATIVE The Broken Stick Problem Taught by Professor Richard C. Larson Mitsui Professor of Engineering Systems and of Civil and Environmental Engineering Segment 1 Hi! My name is Dick Larson

More information

The Derivative of a Function

The Derivative of a Function The Derivative of a Function James K Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 1, 2017 Outline A Basic Evolutionary Model The Next Generation

More information

Stochastic Processes

Stochastic Processes qmc082.tex. Version of 30 September 2010. Lecture Notes on Quantum Mechanics No. 8 R. B. Griffiths References: Stochastic Processes CQT = R. B. Griffiths, Consistent Quantum Theory (Cambridge, 2002) DeGroot

More information

Chapter 2 Descriptive Statistics

Chapter 2 Descriptive Statistics Chapter 2 Descriptive Statistics The Mean "When she told me I was average, she was just being mean". The mean is probably the most often used parameter or statistic used to describe the central tendency

More information

Michigan Farm Bureau Agriscience Lessons -- Connections to Michigan Content Standards

Michigan Farm Bureau Agriscience Lessons -- Connections to Michigan Content Standards Michigan Farm Bureau Agriscience Lessons -- Connections to Michigan Content Standards 2nd GRADE LESSON - "Understanding Insects as Friends or Foes" Michigan Farm Bureau Promotion and Education This lesson

More information

Spider Monkey s Question

Spider Monkey s Question Spider Monkey s Question A Reading A Z Level O Leveled Reader Word Count: 1,108 LEVELED READER O Written by Julie Harding Illustrated by Maria Voris Visit www.readinga-z.com for thousands of books and

More information

Central Limit Theorem and the Law of Large Numbers Class 6, Jeremy Orloff and Jonathan Bloom

Central Limit Theorem and the Law of Large Numbers Class 6, Jeremy Orloff and Jonathan Bloom Central Limit Theorem and the Law of Large Numbers Class 6, 8.5 Jeremy Orloff and Jonathan Bloom Learning Goals. Understand the statement of the law of large numbers. 2. Understand the statement of the

More information

The Shape, Center and Spread of a Normal Distribution - Basic

The Shape, Center and Spread of a Normal Distribution - Basic The Shape, Center and Spread of a Normal Distribution - Basic Brenda Meery, (BrendaM) Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version

More information