Remember your SOCS! S: O: C: S:

Remember your SOCS! S: O: C: S:

1.1: Displaying Distributions with Graphs Dotplot: Age of your fathers Low scale: 45 High scale: 75 Doesn t have to start at zero, just cover the range of the data Label the axis

Stemplot: Age of your father Steps:

Stemplot details Since each stem is a class in the histogram, it looks like a. Benefit: Variations: Round the data so that the final digit is suitable as a leaf. (Ex: 3.468 3.5, 2.567 2.6) You can to double the number of stems when all the leaves would otherwise fall on just a few stems. (Leaves 0-4 go on upper stem, leaves 5-9 go on the lower stem) Ex: Data Set: 110 111 111 113 114 114 114 116 119

Literacy in Islamic Countries

More stemplot Back to back stemplot: Quiz 1 Quiz 2 33 1 58 997650 2 2367778888999 5211 3 234468 9999888775320 4 0112236 00000 5 00

Pie Chart/Bar Graph of Radio Stations by Format

Do you listen while you walk? What is the trend with the use of the MP3 player? You must always look carefully at... ALWAYS think about...

Histogram by hand 1. Divide into classes of equal width. Table 1.3 (p.49): 81-145 Range: 75-155 Specify classes precisely so that each observation falls into exactly. 2. Count # of observations in each class ( ) 3. Draw histogram Horizontal = Vertical = Class Count/Freq 75-84 2 85-94 3 95-104 10 105-114 16 115-124 13 125-134 10 135-144 5 145-154 1

Histograms by TI83/84 (p.59) Calculator steps:

No right choice There are several ways of constructing classes in a histogram. will not give a good idea of the shape of the distribution. Use your judgment! Make sure the classes.

Dealing with Outliers Don t just! You should search for an explanation for an outlier if you find one. Can you get rid of the outlier as bad data or can you live with the statistical consequences of including it?

Examples of things that are symmetric? SYMMETRIC: RIGHT-TAILED SKEWED:

Ogives (relative cumulative frequency graph) p.60 Steps:

Uses

Plots each observation against. Connect points with lines. Vertical axis: Horizontal axis: Remember to look for overall or from the pattern Time plots

Words that need BACK-UP in AP Stats Outlier Skewed Normal Lurking variables Confounding Range Bias...You can always clarify these words!

(a) Write a few sentences to describe what this plot reveals. (b) There is a small peak in the middle of the plot that doesn t fit the overall pattern. Explain this blip. 1) Here is a back-to-back stemplot of the pulse rates of female and male students in one AP Statistics class. Write a few sentences comparing the two distributions. Females Males 0 10 75431 9 0002 8864200 8 04688 88620 7 024578 742 6 00234679 5 5 488 4 8 2) Here is a time plot from buzz.yahoo.com that shows the (illegal) downloading of music using the peer-to-peer software LimeWire during the period May 14 to August 6, 2006.

1.2 Describing Distributions with Numbers How much is a house worth? Manhattan, Kansas, is sometimes called the little apple to distinguish it from the other Manhattan. A few years ago, a house there appeared in the county appraiser s records at $200, 059,000 (true value: $59,500). Before the error was discovered, the county, city, and school board had based their budgets on the total appraised value of real estate, which the one outlier jacked up by 6.5%.

Mean & Median Mean: Median:

Mean/Mean (Centers) Both measure center in different ways, but both are useful. Use median if you want: Mean = Mean/Median of a symmetric distribution are. If a distribution is exactly symmetric,. In a skewed distribution,.

Male/Female Surgeons (# of hysterectomies performed) Put in ascending order (male dr s): odd # 20 25 25 27 28 31 33 34 36 37 44 50 59 85 86 Put in ascending order (female dr s): even # 5 7 10 14 18 19 25 29 31 33

Measures of Spread Range Quartiles Percentiles 5 # Summary Variance Standard Deviation Range = Better measure of spread:

Quartiles and 5 # Summary Steps to calculate quartiles: 5 # Summary:

A modified boxplot:

Boxplots You can see that female dr s perform less hysterectomies than male doctors. Also, there is less variation among female doctors.

Notes on boxplots Best used for of more than 1 distribution. than histograms or stem plots. Always include:

Interquartile Range (IQR) IQR = Measures the spread of the middle ½ of the data. The Rule for Outliers: An observation is an outlier if: Less than or Greater than

Looking at the spread. IQR shows spread of Spacing of the quartiles and extremes about the median give an indication of the of the distribution. Symmetric distributions: 1 st /3 rd quartiles equally distant from the median. In right-skewed distributions: 3 rd quartile will be farther above the median than the 1 st quartile is below it.

Travel Times to Work #1 How long does it take you to get from home to school? Here are the travel times from home to work in minutes for 15 workers in North Carolina, chosen at random by the Census Bureau: 30 20 10 40 25 20 10 60 15 40 5 30 12 10 10

The distribution Describe: Is the longest travel time (60 minutes) an outlier? How many of the travel times are larger than the mean? If you leave out the longest time, how does that change the mean? The mean in this example is because it is sensitive to the influence of extreme observations.

You do: Travel Times to Work #2 Travel times to work in New York State are (on the average) longer than in North Carolina. Here are the travel times in minutes of 20 randomly chosen New York workers: 10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Got friends? Is there a difference between the number of programmed telephone numbers in girls cell phones and the number of programmed numbers in boys cell phones? Do you think there is a difference? If so, in what direction? 1) Count the number of programmed telephone numbers in your cell phone and write the total and M/F on your post-it and pass it up. 2) Make a back-to-back stemplot of this information, then draw boxplots. When you test for outliers, how many do you find for males and how many do you find for females using the 1.5 X IQR test? 3) Find the 5# Summary for each group. Compare the two distributions (SOCS!). 4) It is important in any study that you have data integrity (the data is reported accurately and truthfully). Do you think this is the case here? Do you see any suspicious observations? Can you think of any reason someone may make up a response or stretch the truth? If you DO see a difference between the two groups, can you suggest a possible reason for this difference? 5) Do you think a study of cell phone programmed numbers for a sophomore algebra class would yield similar results? Why or why not?

Draw a histogram for the amount of sleep a class got last night: 6 7 9.5 9 6 4.5 10 8 6 7 7 7 7 7 8 7 8 8.5 9 8.5 7 5 8 6 9 8 6 8 8 4 6 6 Construct a dotplot then find the mean, median and mode for the number of AP classes a class of students are taking this year: 3 4 3 6 5 3 4 4 3 1 3 3 1 1 2 2 2 1 5 5 3 3 2 3 2 2 3 Find the five-number summary, draw a boxplot, and find any outliers for the time the students spent on the internet yesterday (min): 30 90 5 60 60 90 4 120 30 90 45 180 180 120 90 60 240 180 45 120 60 0 180 60 30 120 30 30 90 180 60 45 360 5 240 240 For all 3 graphs, comment on the center, shape, and spread, and prove whether or not there are any outliers.

Section 1.2 Part II... Standard Deviation: Standard deviation looks at. It s the natural measure of for the Normal distribution We like instead of (variance) since the units of measurement are easier to work with (original scale) is the average of the squares of the deviations of the observations from their mean.

Etc s, like the mean,. A few outliers can make s very large. Skewed distributions with a few observations in the single long tail =. ( S is therefore not very helpful in this case) As the observations become more spread about the mean,.

Mean vs. Median Standard Deviation vs. 5# Summary The mean (x-bar) and standard deviation (s) are than the five number summary (min, Q1, med, Q3, max) as a measure of center and spread. No single # describes the spread well. Remember: A graph gives the best overall picture of a distribution. ALWAYS! The choice of mean/median depends upon. When dealing with a skewed distribution,. When dealing with reasonably symmetric distributions,.

S and S^2 S = S^2 = The variance and standard deviation are LARGE if SMALL if

Degrees of Freedom (n-1) Definition: Calculated from the. They are a measure of the amount of information from the sample data that has been used up. Every time a statistic is calculated from a sample, one degree of freedom is used up. If the mean of 4 numbers is 250, we have degrees of freedom (4-1) = 3. Why? mean = 250

Properties of Standard Deviation 1. 2. 3. Choosing a Summary:

A person s metabolic rate is the rate at which the body consumes energy. Metabolic rate is important in studies of weight gain, dieting, and exercise. Here are the metabolic rates of 7 men who took part in a study of dieting: 1792 1666 1362 1614 1460 1867 1439 Find the mean List 1: Observations (x) List 2: Deviations (L1-mean) List 3: Squared deviations (L2)^2 (Sum L3) / (n-1) Calc:

You do! (Long Way Round) Let X = { 3,7,15,23} What is the variance and standard deviation?

You do! (using 1 Var Stats) During the years 1929-1939 of the Great Depression, the weekly average hours worked in manufacturing jobs were 45, 43, 41, 39, 39, 35, 37, 40, 39, 36, and 37. What is the variance and standard deviation?

Linear Transformations A linear transformation changes: Effect of a linear transformation...

Miami Heat Salaries 1) Suppose that each member receives a $100,000 bonus. How will this effect the center, shape, and spread? 2) Suppose that each player is offered 10% increase in base salary. What happened to the centers and spread? Player Salary Shaq 27.7 Eddie Jones 13.46 Wade 2.83 Jones 2.5 Doleac 2.4 Butler 1.2 Wright 1.15 Woods 1.13 Laettner 1.10 Smith 1.10 Anderson.87 Dooling.75 Wang.75 Haslem.62 Mourning.33

Where do I stand? A student gets a test back with a score of 78 marked clearly at the top. A middle-aged man goes to his doctor to have his cholesterol checked. His total cholesterol reading is 210 mg/dl. An employee in a large company earns an annual salary of $42,000. A 10th grader scores 46 on the PSAT Writing test.

Big Idea! You can describe where an individual score falls within a distribution by describing that score s location relative to the mean or median. measure location relative to the median. We use to measure location relative to the mean.

2.1: Measures of Relative Standing and Density Curves A standardized z-score = A z-score is. The absolute value of z tells you how many the score is from the. The sign (positive or negative) of z tells you. Z scores give you the ability to values across distributions with different means and standard deviations.

Jenny scored an 86 on her first stats test. How did she perform among her classmates? 1) Look at distribution Outliers? Shape? 2) Summary Stats 79 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 83 89 84 82 77 72

1) Jenny scored above average. But by how much? 2) Katie scored the highest, 93. What is her z-score? What does it mean? 3) Norman got a 72. what is his z-score? What does it mean?

Percentiles Norman got a 72 on his exam. Only one person did worse than he did out of a total of 25 people. What is his percentile? Katie got the highest score out of the class (she was the 93). What is her percentile?

On an index card, write your height in inches, then write your height on the board. Hold up your index card and put yourselves in order around the room (shortest to tallest). Count the number of people who are shorter than you (include yourself). Calculate the mean, standard deviation, 5 # summary. Calculate your percentile, then find how many standard deviations you are above or below the mean (find your z-score). Write your percentile and z-score on the back of your index card, and hold it up when Ms. S. tells you to. Look around the room. Does this make sense?

Chebyshev s Inequality: You can use this inequality for (normal or skewed). Describes the of observations in any distribution that fall within a specified number of standard deviations of the mean.

Strategy for exploring data on a single quantitative variable: 1. Graph it 2. Overall pattern? Striking deviations? 3. Numerical summary to describe center/spread? 4. Describe pattern w/smooth curve if it s regular =

Density Curve Example Distribution Symmetric Both tails from No gaps/obvious Smooth curve = Curve is a for the distribution (ignores irregularities and outliers)

From histogram to density curve

Why a smooth curve? Histogram depends on our choice of classes, but when we use a curve, it doesn t depend on any choices we make (easier to work with) Use a smooth curve to describe what of the observations fall in each range of values, not the of the observations. Our eyes respond to the areas of the bars in a histogram. Same is true of a smooth curve: We adjust the scale of the graph so the total area under the curve =.

A density curve is a curve that: - - Important Points. 1. The curve doesn t! 2. It is an description of the data an approximation but is accurate enough for practical use (no real set of data is exactly described by a density curve) 3. Foundation for!

Example 2.5: Reading d.c. s Skewed slightly Shaded area: 7-8 Area under the curve = Therefore, % of all from this distribution have values between 7 and 8. * The real power of d.c. s with normal distributions = based on curve => inference.

Density Curves have many shapes. Left: The median and mean of a symmetric density curve are. Right: The median and mean of a right-skewed density curve are (mean pulled towards tail).

Since areas under a density curve represent proportions of the total # of observations Median of a density curve is the, the point with % of the area under the curve to its left, and the remaining % of the area to the right. divide the area under the curve into quarters (25% of the area under the curve is to the left of Q1 )

Mean of a density curve The mean is the point at which the curve would balance if it were made of solid material. The! Look at figure 2.7 on page 127

When does Mean = Median? The median and the mean are the same for a. They both lie at the of the curve. The mean of a skewed curve is pulled away from the median in the direction of the.

Notation Mean and standard deviation for actual observations (samples): Mean and standard deviation for idealized distributions (populations):

Example: A density curve consists of a straight line drawn from the origin (0,0); the slope is 1. a) Find the point of termination for this line (hint: use the fact that this is a valid density curve). b) Find Q1, Q2, Q3 c) Relative to the median, where would you expect the mean of the distribution to lie? d) What percentage of the data lies below.5? What percentage of the data lies above 1.5?

2.2: Normal Distributions Note on Uniform Distributions

3 Reasons why we like Normal Distributions Good of real data (ex: SATs, psychological tests, characteristics of populations ) Good to results of many kinds of chance outcomes. Many work well for roughly symmetrical distributions. Many data sets tend to be (characteristics of biological populations) TI83: student heights, L1, graph

Normal Distributions Described by giving its mean and std. deviation controls the spread of a normal curve. Figure shows curve w/different values of. Changing w/o changing moves the curve along the horizontal axis w/o changing spread.

Locating the standard deviation by eyeballing the curve: As we move out in either direction from the center changes from falling ever more steeply µ, the curve

The 68-95-99.7 Rule States: Common Properties of Normal Curves: They all have (where change of curvature takes place). only provides an approximate value for the proportion of observations that fall within 1, 2, or 3 std. devs of the mean. σ σ µ µ

Example #1 Suppose that taxicabs in NYC are driven an average of 75,000 miles per year with a standard deviation of 12,000 miles. What information does the empirical rule tell us?

2 Normal curves What do you notice about their means? What do you notice about their standard deviations?

Standard Normal Table - A Table A is a table of (proportions/probabilities) under the standard Normal curve. The table entry for each value z is the under the curve to the of z.

Steps for solving problems with Normal Distributions: 1. 2. 3. 4.

Finding Areas to the Left Find the proportion of observations from the standard normal distribution that are less than 2.22. That is: Find the probability that z is less than 2.22 or P (z < 2.22) =

Finding Areas to the Right Find the proportion of observations from the standard normal distribution that are greater than -2.15. That is: find P (z > -2.15)

Table A Practice Use Table A to find the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. 1) Z is less than or equal to -2.25 2) Z is greater than or equal to -2.25 3) Z > 1.77 4) -2.25 < z < 1.77

Example The mean of women is 64.5 inches, and the standard deviation is 2.5 inches. What proportion of all young women are less than 68 inches tall?

Example The level of cholesterol in the blood is important because high cholesterol levels may increase the risk of heart disease. The distribution of blood cholesterol levels in a large population of people of the same age and sex is roughly normal. For 14 year old boys, the mean is 170 mg/dl and the 2 standard deviation is 30 mg/ dl. Levels above 240 mg/dl may require medical attention. What percent of 14-year-old boys have more than 240 mg/dl of cholesterol?

What percent of 14 year old boys have between 170 and 240 mg/dl?

Finding a value given a proportion Use Table A backwards! 1) Find the given proportion in the of the table 2) Read the corresponding 3) Unstandardize to get the observed (x) value. Voila!

Example Scores on the SAT verbal test in recent years follow approximately the N(505,110) distribution. How high must a student score in order to place in the top 10% of all students taking the SAT?

Special Note. X is greater than x is greater than or equal to because it is a curve. That is, there is where x = 240. There may be a boy with an exact cholesterol level of 240, but. The normal distribution is therefore an not a description of every detail in the exact data.

Normal Probability Plot If the points on a Normal Probability Plot make a than the data are. Use Calculator Don t overreact to minor wiggles in the plot Normality cannot be assumed if there is skewness or outliers (don t use Normal distribution if these things occur)!