The science of learning from data.

Transcription

1 STATISTICS (PART 1)

2 The science of learning from data. Numerical facts Collection of methods for planning experiments, obtaining data and organizing, analyzing, interpreting and drawing the conclusions or making a decision

3 Population: A collection, or set, of individuals or objects or events whose properties are to be analyzed. Sample: A subset of the population. Population Sample Element: Entities on which data are collected. Observation: Value of variable for an element. Data Set: A collection of observation on one or more variables.

4 Grouped data Data that has been organized into groups (into a frequency distribution). Data Frequency Ungrouped data - Data that has not been organized into groups. Also called as raw data. Data Frequency

5 VARIABLES QUALITATIVE QUANTITATIVE NOMINAL Example: gender, color ORDINAL Example: Pass/Fail, Good, Bad DISCRETE Example: Counts- number of items/integers CONTINUOUS Example: Measurement- Length, weight

6 Identify each of the following examples as qualitative or quantitative variables. 1. The residence hall for each student in a statistics class. (qualitative 2. The amount of gasoline pumped by the next 10 customers at the local Unimart. (quantitative ) 3. The amount of radon in the basement of each of 25 homes in a new development. (quantitative ) 4. The color of the baseball cap worn by each of 20 students. (qualitative) 5. The length of time to complete a mathematics homework assignment. (quantitative ) 6. The state in which each truck is registered when stopped and inspected at a weigh station. (qualitative

7 Discrete data is data which can only take certain values, or can be counted. The number of people in a room can only be 1, 2, 3, and not 1.23, 1.57, Example: - Number of car on a road - Number of children in a family - The shoe sizes of students in a class Continuous data cannot assume exact values but can assume any values between two given values. The data is acquired through the process of measuring. For example, the height 175 cm (correct to the nearest cm) could have arisen from any values in the range. - Weight of people - Speeds of motor boats at a particular part of a race - The times taken by each of student to run 100m

8 Provide simple summaries about the sample and the measures STATISTICS Trying to reach conclusion that extend beyond the immediate data alone Descriptive Inferential - Measurement of central tendency - Measurement of dispersion - T-test - Analysis of Variance (ANOVA) - Analysis of Covariance (ANCOVA) - Regression analysis

9 Descriptive Statistics A study on data summary or describes a collection, data organization (presentation of data in a more informative way such as graphical, diagrams and charts). In general divided by two categories :- - Data presentation (display) - Tabular - Charts/graphs 9

10 Inferential Statistics Branch of statistics: using a sample to draw conclusions about a population (basic tool: probability). Consists of generalizing from samples to population, performing estimations and hypothesis tests, determining relationships among variables, and making predictions. Area statistics which are deal with decision making procedures. Population consists of all subjects (human or otherwise) that are being studied. Sample is a group of subjects selected from a population. 10

11 Tabular presentation for qualitative data is usually in the form of frequency table that is a table represents the number of times the observation occurs in the data. *Qualitative :- characteristic being studied is nonnumeric. Examples:- gender, religious affiliation or eye color. The most popular charts for qualitative data are: 1. bar chart/column chart; 2. pie chart; and 3. line chart.

12 Types of Graph Qualitative Data

13 Example 6.2: Frequency Table Example 6.3: Observation Frequency Malay 33 Chinese 9 Indian 6 Others 2 Bar Chart: used to display the frequency distribution in the graphical form.

14 Example 6.4 : Pie Chart: used to display the frequency distribution. It displays the ratio of the observations Malay Chinese Indian Others Example 6.5: Line chart: used to display the trend of observations. It is a very popular display for the data which represent time. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

15 Tabular presentation for quantitative data is usually in the form of frequency distribution that is a table represent the frequency of the observation that fall inside some specific classes (intervals). *Quantitative : variable studied are numerically. Examples:- balanced in accounts, ages of students, the life of an automobiles batteries such as 42 months). Frequency distribution: A grouping of data into mutually exclusive classes showing the number of observations in each class. There are few graphs available for the graphical presentation of the quantitative data. The most popular graphs are: 1. histogram; 2. frequency polygon; and 3. ogive.

16 Example 6.5: Frequency Distribution Weight (Rounded decimal point) Frequency Example 6.6: Histogram: Looks like the bar chart except that the horizontal axis represent the data which is quantitative in nature. There is no gap between the bars.

17 Example 6.7 : Frequency Polygon: looks like the line chart except that the horizontal axis represent the class mark of the data which is quantitative in nature. Example 6.8 : Ogive: line graph with the horizontal axis represent the upper limit of the class interval while the vertical axis represent the cummulative frequencies.

18 Constructing Frequency Distribution When summarizing large quantities of raw data, it is often useful to distribute the data into classes. Weight Frequency Total 100 Weight of 100 male students in XYZ university A frequency distribution for quantitative data lists all the classes and the number of values that belong to each class.

19 For quantitative data, an interval that includes all the values that fall within two numbers; the lower and upper class which is called class. Class is in first column for frequency distribution table. *Classes always represent a variable, non-overlapping; each value is belong to one and only one class. The numbers listed in second column are called frequencies, which gives the number of values that belong to different classes. Frequencies denoted by f. Variable Third class (Interval Class) Table 6.1: Weight of 100 male students in XYZ university Weight Frequency Total 100 Frequency column Frequency of the third class.

20 The class boundary is given by the midpoint of the upper limit of one class and the lower limit of the next class. The difference between the two boundaries of a class gives the class width; also called class size. Formula: - Class Midpoint or Mark Class midpoint or mark = (Lower Limit + Upper Limit)/2 - Finding The Number of Classes Number of classes, i = 1 3.3log n - Finding Class Width For Interval Class class width, c = (Largest value Smallest value)/number of classes

21 Cumulative Frequency Distributions A cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class. In cumulative frequency distribution table, each class has the same lower limit but a different upper limit. Table 6.2: Class Limit, Class Boundaries, Class Width, Cumulative Frequency Weight (Class Interval) Number of Students, f Class Boundaries TOTAL 100 Cumulative Frequency = = = = 100

22 Example 6.9: From Table 6.1: Class Boundary Weight (Class Interval) Class Boundary Frequency Total 100

23 The diameters, in mm, of 20 pipes is as follows: a) How many class interval? b) Build a frequency distribution table. c) What is the lower boundary for the first class?

24 Answer: a) Number of class 1 3.3log c b) Diameter, mm Class (Class Interval) Boundary Frequency Total 20

25 Measures of Central Tendency - Mean - Median - Mode Measures of average are also called measures of central tendency and include the mean, median, mode, and midrange. Measures of Dispersion - Variance - Standard deviation After know about average, you must know how the data values are dispersed. That is, do the data values cluster around the mean.

26 Mean Mean of a sample is the sum of the sample data divided by the total number sample. UNGROUPED DATA: x sum of observations number of observations x x1 x2... x n n n GROUPED DATA: When the data has been grouped into intervals and the mid-points of the intervals are denoted by x i, x sum of observations number of observations fx f x f x f x f f n n

27 Given a data set: 3, 2, 4, 2, 6, 8, 10, 5. Find the average. (Answer: 5) Consider data set of weights of 30 students. Find the mean. Weight (x) Frequency (f) Answer: kg Consider data set of weights of 30 students. Find the mean of grouped data. Weight(kg) Frequency (f) f 30 Answer: 46.5 kg

28 Median The median is the middle value of a set of numbers arranged in order of magnitude and normally is denoted by, x UNGROUPED DATA: The median depends on the number of values n. If n is odd, then the median is the n 1 2 th value. But if n is even, then the median is arithmetic mean of the n 2 th value and the n 1 2 th value. GROUPED DATA: The median of frequency distribution data can be described as: f Fj 1 x L c 2 where f j L = the lower class boundary of the median class c = the size of median class interval F f j1 j the sum of frequencies of all classes lower than the median class the frequency of the median class

29 Find the median of the following data: 1. 4,6,3,1,2,5,7,3 (Answer: 3.5) 2. 8,3,4,1,7,8,9,5,3 (Answer: 5) 3. Class Frequency Total 30 Answer: Class interval Frequency Answer: 11

30 Mode The mode of a set of numbers is the value which occurs most often and denoted by x, UNGROUPED DATA: The mode of ungrouped data can be defined as the value which occurs most often. The mode has the advantage in that it is easy to calculate and eliminates the effect of extreme values. Note: - If a set of data has 2 measurements with higher frequency, therefore the measurements are assumed as data mode. - If a set of data has more than 2 measurements with higher frequency so the data can be assumed as no mode. GROUPED DATA: The median of frequency distribution data can be described as: 1 xˆ L c 1 2 where L c 1 2 the lower class boundary of the modal class = the size of the modal class interval the difference between the modal class frequency and the class before it the difference between the modal class frequency and the class after it

31 Find the mode of the following data: 1. 3,4,1,3,8,2,9,3,2 (Answer: 3) 2. 2,8,1,2,9,10,8,11 (Answer: 2 and 8) 3. 3,6,7,6,3,7 (Answer: no mode) 4. Class Frequency Total 30 Answer: Class interval Frequency Answer: 14.64

32 When the mean, median and mode are all equal, the distribution of the data set has a bell-shaped curve. The distribution is then said to be symmetric. If Mode < Median < Mean, then the distribution is said to be positive/right skewed, meaning there are a few unusual large values. If Mean < Median < Mode, then the distribution is said to be negative/left skewed, that is there are some unusual small values.

33 The standard deviation from the mean is used widely in statistics to indicate the measure of dispersion. Small standard deviation tells that most of the data is close to the mean. While large standard deviation shows that much of the data is far from the mean. Small SD Large SD Mean Mean

34 UNGROUPED DATA: Variance 1 Standard deviation 1 x x s n x x s n GROUPED DATA: Variance 1 Standard deviation 1 fx n x s f fx n x s f

35 Example 6.10 (Grouped data) Find the variance and standard deviation of the sample data below: Weight (Class Interval) Frequency, f Class Mark, x fx Cumulative Frequency, F Class Boundary 2 fx fx Total S 2 fx 2 f 1 2 n x? S fx 2 f 1 2 n x? Answer : s 2 =8.61;s=2.93

36 (a) The marks of an examination are given by the following data: 12, 27, 13, 21, 36, 56, 53, 55, 59, 83, 92, 75, 67, 80, 91, 99, 84, and 77. Evaluate the variance and standard deviation for this data. (b) Consider data set of weights of 30 students. Find the standard deviation. Weight(kg) Frequency (f) Answer: (a) (b) s s , s