Chapter 3 Statistics for Describing, Exploring, and Comparing Data

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1 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

2 Review Chapter 1 Statistical and critical thinking, types of data, collecting data. Chapter 2 Frequency distributions, summarizing data with graphs.

3 Preview Descriptive Statistics In this chapter we ll learn to summarize or describe the important characteristics of a data set (mean, standard deviation, etc.). Inferential Statistics In later chapters we ll learn to use sample data to make inferences or generalizations about a population.

4 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

5 Part 1 Basics Concepts of Measures of Center Measure of Center

6 1. Arithmetic Mean x is pronounced x-bar and denotes the arithmetic mean of a sample. Let n be number of data values in the sample, Simply call it mean. x x n is pronounced mu and denotes the arithmetic mean of a population. Let N be the number of data values in the population x N Mean is also called, by some people, average.

7 Advantages 1. Mean Tend to vary less than other measures of center Takes every data value into account Disadvantage One extreme value can affect it dramatically; is not a resistant measure of center

8 Example 1 - Mean Table 3-1 includes counts of chocolate chips in different cookies. Find the mean of the first five counts for Chips in Ahoy regular cookies: 22 chips, 22 chips, 26 chips, 24 chips, and 23 chips chips

9 Median 2. Median the middle value when the original data values are sorted. often denoted by x (pronounced x-tilde ) not affected by an extreme value - is a resistant measure of the center

10 Median Odd Number of Values First sort the values. Case 1, the number of values is odd Sort in order: (in order - odd number of values) Median is 0.73

11 Median Even Number of Values First sort the values. Case 2, the number of values is even Sort in order: Median is 0.915

12 3. Mode Mode the value that occurs with the greatest frequency Data set can have one, more than one, or no mode Mode is the only measure of central tendency that can be used with nominal data.

13 Mode - Examples a Mode is 1.10 b Bimodal - 27 & 55 c No Mode

14 4. Midrange Midrange = maximum value + minimum value 2

15 Midrange Sensitive to extremes Rarely used Redeeming Features (1) Easy to compute (2) Shows there are several ways to define the center

16 Round-off Rule for Measures of Center Carry one more decimal place than is present in the original set of values.

17 Critical Thinking Think about whether the results are reasonable. Think about the method used to collect the sample data.

18 Example In the following examples, why the mean and median would not be meaningful statistics? a. Rank (by sales) of selected statistics textbooks: 1, 4, 3, 2, 15 b. Numbers on the jerseys for the New Orleans Saints: 12, 74, 77, 76, 73, 78, 88, 19, 9, 23, 25

19 Part 2 Beyond the Basics of Measures of Center

20 Calculating a Mean from a Frequency Distribution To find the mean, use class midpoint of classes for variable. x ( f x) f

21 Example Estimate the mean from the IQ scores in Chapter 2. x ( f x) f

22 Weighted Mean When data values are assigned different weights, w, we can compute a weighted mean. x ( wx) w

23 Example Weighted Mean In her first semester of college, a student of the author took five courses. Her final grades along with the number of credits for each course were A (3 credits), A (4 credits), B (3 credits), C (3 credits), and F (1 credit). The grading system assigns quality points to letter grades as follows: A = 4; B = 3; C = 2; D = 1; F = 0. Compute her grade point average. Solution Use the numbers of credits as the weights: w = 3, 4, 3, 3, 1. Replace the letters grades of A, A, B, C, and F with the corresponding quality points: x = 4, 4, 3, 2, GPA

24 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

25 Key Concept Discuss measures of variation, such as standard deviation, for analyzing data. Make understanding and interpreting the standard deviation a priority.

26 Part 1 Basics Concepts of Variation

27 Definition 1. Range Range = (maximum value) (minimum value) It is very sensitive to extreme values; Not as useful as other measures of variation.

28 Round-Off Rule for Measures of Variation Same as in the case of measure of center. Round only the final answer.

29 Sample Standard Deviation Formula 2. Standard deviation The standard deviation of a set of sample values, denoted by s, is a measure of how much data values deviate away from the mean. s ( xx) 2 n 1

30 Sample Standard Deviation (Shortcut Formula) s 2 2 n x ( x) nn ( 1)

31 Standard Deviation Important Properties s is usually positive (it is never negative). s increase dramatically with the inclusion of outlier(s). The units of s are the same as the units of the original data values.

32 Example Find the standard deviation of these numbers of chocolate chips: 22, 22, 26, 24

33 Example s x x n

34 Range Rule of Thumb for Understanding Standard Deviation Find rough estimates of the minimum and maximum usual sample values as follows: Minimum usual value = mean 2(standard deviation) Maximum usual value = mean + 2(standard deviation)

35 Example Using the 40 chocolate chip counts for the Chips Ahoy cookies, the mean is 24.0 chips and the standard deviation is 2.6 chips. Use the range rule of thumb to find the minimum and maximum usual numbers of chips. Minimum usual value = 18.8 Maximum usual value = 29.2 Would a cookie with 30 chocolate chips be unusual? Yes.

36 Comparing Variation in Different Samples Two sample means are approximately the same, and the units are the same, then compare two sample standard deviations. If two sample means are very different, or units are different, then use coefficients of variance to compare.

37 Coefficient of Variation The coefficient of variation (or CV) for a set of nonnegative sample or population data, Sample Population cv s 100% cv 100% x

38 Coefficient of Variation Example 8. Cookies have average 24.0 chips and standard deviation is s = 2.6 chips. Coke has x = lb, and s = lb. Compare the variation using CV. # of chocolate chips: CV = 10.8% Weight of coke: CV = 0.9%

39 Population Standard Deviation ( x ) 2 N similar to the sample s formula Here N is used instead of n 1.

40 Variance - Notation s = sample standard deviation s 2 = sample variance = population standard deviation 2 = population variance

41 Part 2 Beyond the Basics of Variation

42 Empirical (or ) Rule For data sets having a distribution that is approximately bell shaped, the following properties apply: About 68% of all values fall within 1 standard deviation of the mean. About 95% of all values fall within 2 standard deviations of the mean. About 99.7% of all values fall within 3 standard deviations of the mean.

43 Chebyshev s Theorem The proportion (or fraction) of any set of data lying within K standard deviations of the mean is always at least 1 1/K 2, where K is any positive number greater than 1. For K = 2, at least 3/4 (or 75%) of all values lie within 2 standard deviations of the mean. For K = 3, at least 8/9 (or 89%) of all values lie within 3 standard deviations of the mean.

44 Example IQ scores have a mean of 100 and a standard deviation of 15. What can we conclude from Chebyshev s theorem? At least 75% of IQ scores are between 70 and 130. At least 88.9% of IQ scores are between 55 and 145.

45 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

46 Key Concept Relative standing compare values from different data sets, compare values within the same data set. The z score. Percentiles and quartiles, as well as boxplot.

47 Part 1 Basics of z Scores, Percentiles, Quartiles, and Boxplots

48 Measures of Position z Score Sample Population x x x z z s Round z scores to 2 decimal places

49 Interpreting Z Scores Ordinary values: Unusual Values: 2 zscore 2 zscore 2 or zscore 2

50 Example The author of the text measured his pulse rate to be 48 beats per minute. Is that pulse rate unusual if the mean adult male pulse rate is 67.3 beats per minute with a standard deviation of 10.3? z xx s 10.3 Not unusual

51 Percentiles are measures of location. There are 99 percentiles denoted P 1, P 2,..., P 99, which divide a set of data into 100 groups.

52 Finding the Percentile of a Data Value Sort data from smallest to largest. number of values less than x Percentile of value x = 100 total number of values Round the percentile.

53 Example For the 40 Chips Ahoy cookies, find the percentile for a cookie with 23 chips. Answer: We see there are 10 cookies with fewer than 23 chips, so Percentileof A cookie with 23 chips is in the 25 th percentile.

54 Converting from the kth Percentile to the Corresponding Data Value L Notation k n n k total number of values in the data set percentile being used 100 L locator that gives the position of a value P k k-th percentile

55 Converting from the kth Percentile to the Corresponding Data Value If L is a decimal number, round up. Still denoted by L. The value corresponding to P k is the L-th data value in the sorted list. If L is a whole number, the value corresponding to P k is the value midway between L-th value and (L+1)-th value in the sorted list. Example: 2, 3, 6, 8, 10, 11, 13, 15, 16, 18 Find value corresponding to P 30 and P 35. 7, 8

56 Quartiles Q 1, Q 2, Q 3 divide sorted data values into four equal parts 25% 25% 25% 25% (minimum) Q 1 Q 2 Q 3 (maximum) (median) Interquartile Range: IQR = Q 3 Q 1

57 5-Number Summary For a set of data, the 5-number summary consists of these five values: 1. Minimum value 2. First quartile Q 1 3. Second quartile Q 2 (median) 4. Third quartile, Q 3 5. Maximum value A boxplot use these 5 numbers.

58 Boxplot - Construction 1. Find the 5-number summary. 2. Construct a scale cover all 5 numers. 3. Construct a box from Q1 to Q3 and draw a line in the box at Q2. 4. Draw lines from the box to the minimum and maximum values.

59 Boxplot - Construction Example. Construct Boxplot using 40 Chips Ahoy cookies. Short cut for finding Q1, Q2 and Q3. Q2 is the median. Q2 separate sorted data into left half and right half. Q1 is the median of values on left half, Q3 the median of values on right half

60 Boxplots - Normal Distribution Normal Distribution: Heights from a Simple Random Sample of Women

61 Boxplots - Skewed Distribution Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches

62 Part 2 Outliers and Modified Boxplots

63 Outliers for Modified Boxplots In modified boxplots, a data value is an outlier if it is: or above Q IQR below Q IQR

64 Modified Boxplot Construction A modified boxplot is constructed with these specifications: A special symbol (such as an asterisk) is used to identify outliers. The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier.

65 Outliers for Modified Boxplots Example. Number of chips in cookies (Hannaford) 13, 15, 16, 21, 15, 14, 14, 15, 13, 13, 16, 11 14, 12, 13, 12, 14, 12, 16, 17, 14, 16, 14, *

66 Modified Boxplots - Example

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