Percentile: Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included:

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1 AP Statistics Chapter 2 Notes 2.1 Describing Location in a Distribution Percentile: The pth percentile of a distribution is the value with p percent of the observations (If your test score places you in the 88 th percentile, then 88 percent of the scores were below yours) Percentiles should be numbers (round to nearest integer if necessary) To find percentile of n in a data set, calculate the percent of the values in the distribution that are n in the data set. If two observations have the same value, they will be at the same. Use percentiles to individual values within a distribution of data. Percent correct on a test does not measure the same thing as a percentile (Think SAT scores) there is no 0 percentile rank - the lowest score is at the 1 st percentile there is no 100th percentile - the highest score is at the 99th percentile. you perform the same mathematical operations on percentiles that you can on raw scores. You cannot, for example, compute the mean of percentile scores, as the results may be misleading. Definition 1: A percentile is a measure that tells us what percent of the total frequency scored at or below that measure. A percentile rank is the percentage of scores that fall at or below a given score. Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is included: Definition 2: A is a measure that tells us what percent of the total frequency scored that measure. A percentile rank is the percentage of scores that fall below a given score. Formula: To find the percentile rank of a score, x, out of a set of n scores, where x is not included: Where B = number of scores below x E = number of scores equal to x n = number of scores See this formula in more detail in the Examples section. Example: If Jason graduated 25 th out of a class of 150 students, then 125 students were ranked below Jason. Jason's percentile rank would be: Example: If Jason graduated 25 th out of a class of 150 students, then 125 students were ranked below Jason. Jason's percentile rank would be: Jason's standing in the class at the 84 th percentile is as higher or higher than 84% of the graduates. Good job, Jason! Jason's standing in the class at the percentile is higher than 83% of the graduates. Good job, Jason! The slight difference in these two definitions can lead to significantly different answers when dealing with small amounts of data.

2 Examples: 1) Karl takes the big Earth Science test and his teacher tells him that he scored at the 92 nd percentile. Is Karl pleased with his performance on the test? 2) Sue takes the Chapter 4 math test. If Sue's score is the same as "the mean" score for the math test. What would be her percentile? 3) If Ty scores at the 75th percentile on the Social Studies test, what percent of the scores were below Ty s? 4) The math test scores were: 50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99. a) Find the percentile rank for a score of 86 on this test. b) Find the percentile for a score of 84. Cumulative Relative Frequency Graphs: -Used to: describe the position of an individual within a distribution or to locate a specified percentile of the distribution. To make a cumulative relative frequency graph ( ogive ), 1) plot a point corresponding to the cumulative relative frequency in each class at the smallest value of the next class. 2) Start a cumulative relative frequency graph with a point at a height of 0% at the smallest value of the first class and the last point at a height of 100%. 3) Connect consecutive points with line segments to complete your graph. 4) Be sure to have and a consistent.

3 Ex: Age of U.S. presidents at inauguration. Age Frequency Step 1: Expand the table to include cumulative relative frequency Age Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency Step 2: Graph: Step 3: Interpreting a cumulative relative frequency graph.

4 -Read THINK ABOUT IT (pg. 89) -CYU Pg.89 Measuring Position: z-scores z-score = standardized value of x Formula: Example: Mr. Pryor s second period consists of 25 students. The mean grade on the first exam was 80, the median was also 80, and the standard deviation was Find the standardized scores (z-scores) for each of the following students. Interpret each value in context. a) Katie, who scored 93. b) Norman, who scored 72 c) Jenny, who scored an 86. d) Jenny earned an 82 on Mr. Goldstone s chemistry test. Mr. Goldstone told the class that the distribution was fairly symmetric with a mean of 76 and a standard deviation of 4. On which test did Jenny perform better relative to the class? Justify your answer.

5 Facts about z-scores A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. Observations larger than the mean have positive z-scores. Observations smaller than the mean have negative z-scores. A z-score of 0 results from an observation that is equal to the mean. A z-score is not measured in the same units as the variable. It just indicates how many standard deviations an observation is above or below the mean. We standardize observations to express them on a common scale. Use z-scores to compare the position of individuals in different distributions. (Distributions must be roughly the same shape) CYU: Pg. 91 Transforming Data: Alt. Example: Test Scores Here are a graph and table of summary statistics for a sample of 30 test scores. The maximum possible score on the test was 50 points. Collection 1 Dot Plot Score n x s x Min Q1 M Q3 Max IQR Range Score

6 Suppose that the teacher was nice and added 5 points to each test score. How would this change the shape, center, and spread of the distribution? (Here are graphs and summary statistics for the original scores and the +5 scores:) Score Score_Plus n x s x Min Q1 M Q3 Max IQR Range Score Score + 5 Suppose that the teacher in the previous alternate example wanted to convert the original test scores to percents. Since the test was out of 50 points, he should multiply each score by 2 to make them out of 100. Here are graphs and summary statistics for the original scores and the doubled scores. Collection 1 Score Scorex n x s x Min Q1 M Q3 Max IQR Range Score Score x 2

7 Effect of Adding (or Subtracting) a Constant Adding the same number a (either positive, zero, or negative) to each observation: Adds a to measure of center and location (mean, median, quartiles, percentiles), but Does not change the shape of the distribution Does not change the measures of spread (range, IQR, standard deviation) Effect of Multiplying (or Dividing) by a Constant Multiplying (or dividing) each observation by the same number b (either positive, zero, or negative): Multiplies (divides) measures of center and location (mean, median, quartiles, percentiles) by b Does not change the shape of the distribution Multiplies (divides) measures of spread (range, IQR, standard deviation) by /b/ **The only way to change the shape of a distribution is by multiplying (or dividing) the values in a distribution by a variable. (more on this in chapter 8) Q: What is the relationship between transformations and z-scores? (Hint: See THINK ABOUT IT. Pg.97) CYU: Pg.97

8 Density Curves 1) Plot your data 2) Look for overall pattern: shape, center, spread departures from pattern Exploring Quantitative Data 3) Calculate a numerical summary to describe center and spread: 4) Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve A density curve is a curve that Is always the horizontal axis, and Has area exactly underneath it. Describes the overall pattern of a distribution The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. Density curves come in many shapes, the most commonly used density curve is the curve. (also t curves and chi-square curves) It is often a good description of the overall pattern of a distribution. No set of real data is exactly described by a density curve. Outliers, which are departures from the overall pattern, are not described by the curve. The curve is an that is easy to use and accurate enough for practical purposes. of a density curve is the equal-areas point. (half the area to the left and right of the point) The mean of a density curve is the point at which the curve would balance if made of solid material The mean and median are the same for a density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

9 CYU: Pg. 103

10 AP Statistics Chapter 2 Notes 2.2 Normal Distributions Normal curves are: A type of curve They describe distributions called Distributions They are, single-peaked,. They are completely described by giving its mean μ and its standard deviation σ. The mean is located at the of the symmetric curve and it s the same as the. Changing μ without changing σ moves the Normal curve along the horizontal axis without changing its spread. The standard deviation controls the of a Normal curve. Curves with standard deviations are more spread out. Normal Distributions: A is described by a Normal. A Normal Distribution is completely specified by two numbers: and. Normal curve is symmetric with the mean at its center. The is the distance from the center to the points on either side. Abbreviation for a Normal Distribution: Examples of Normal Distributions: -scores on tests taken by many people (SAT, ACT, EOC, and IQ tests) -repeated careful measurements of the same quantity (measuring diameters of tennis balls) -characteristics of biological populations (length of crickets, yields of corn, weight of tuna) **Not all distributions are Normal! (Example: income)

11 The Rule: Notes: Examples: The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7 th grade students in Gary, Indiana, is close to Normal. Suppose that the distribution is exactly Normal with mean µ = 6.84 and standard deviation σ = 1.55 a) Sketch a Normal density curve for this distribution of test scores. Label the points that are one, two and three standard deviations from the mean. b) What percent of the ITBS vocabulary scores are less than 3.74? c) What percent of the scores are between 5.29 and 9.94? Show your work.

12 Alt. Example: Batting Averages The histogram below shows the distribution of batting average (proportion of hits) for the 432 Major League Baseball players with at least 100 plate appearances in the 2009 season. The smooth curve shows the overall shape of the distribution. The mean of the 432 batting averages was with a standard deviation of Suppose that the distribution is exactly Normal with = and = (a) Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean. (b) What percent of the batting averages are above 0.329? Show your work. (c) What percent of the batting averages are between and.0.227? Show your work. (d) What percent of batting averages are above 0.261?

13 CYU. Pg.114 The Standard Normal Distribution: The Standard Normal Distribution is the Normal Distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution, then the standardized variable z has the standard Normal distribution. Examples: Finding areas under the standard Normal curve: Find the proportion of observations from the standard normal curve that are: a) Less than b) More than 0.82 c) Between and 0.81 (Always draw a sketch of the standard normal curve!)

14 Working backwards: In a standard Normal distribution, 20% of the observations are above what value? Example: Tiger on the Range. (STATE, PLAN, DO, CONCLUDE) On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When Tiger hits his driver, the distance the ball travels follows a Normal distribution with mean 304 yards and standard deviation 8 yards. a) What percent of Tiger s drives travel at least 290 yards? b) What percent of Tiger s drives travel between 305 and 325 yards? Q) What if our z does not show on table A? See Ex: Cholesterol in Young Boys. Pg. 122

15 CYU: Pg.124 Assessing Normality: The procedures presented in this section apply only to Normal distributions. If a distribution is not Normal to begin with, then standardizing it will not make it normal and then our calculations of probability (area under the normal curve) will be inaccurate. We need to establish the shape of the distribution first before we use our Normal distribution techniques. Steps to assess Normality: 1) : Make a dotplot, stemplot, or histogram. Check to see if the graph is approximately symmetric, single-peaked/bell-shaped. 2) Check whether the data follows the rule. 3) Make a : If the points lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non- Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot. Example: Making a Normal Probability Plot: (see pg.133 #63 or 64) Sharks: Here are the lengths in feet of 44 great white sharks: (a) Enter these data into your calculator and make a histogram. Then calculate one-variable statistics. Describe the shape, center, and spread of the distribution of shark lengths.

16 (b) Calculate the percent of observations that fall within one, two, and three standard deviations of the mean. How do these results compare with the rule? (c) Use your calculator to construct a Normal probability plot. Interpret this plot. (d) Having inspected the data from several different perspectives, do you think these data are approximately Normal? Write a brief summary of your assessment that combines your findings from (a) through (c).