MAT 155. Z score. August 31, S3.4o3 Measures of Relative Standing and Boxplots

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Relative Standing and Boxplots Key Concept Measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set, can be used to compare

1 MAT 155 Dr. Claude Moore Cape Fear Community College 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 Key Concept Measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set, can be used to compare values from different data sets, or to compare values within the same data set. The most important concept is the z score. We will also discuss percentiles and quartiles, as well as a new statistical graph called the boxplot. Z score Interpreting Z Scores z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean. Sample Population Round z scores to 2 decimal places Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: 2 z score 2 Unusual Values: z score < 2 or z score > 2 1

Percentiles are measures of location. There are 99 percentiles denoted P1, P2,... P99, which divide a set of data into 100 groups with about 1% of the values in each group.

Converting from the kth Percentile to the Corresponding Data Value Quartiles Are measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the

2 Percentiles are measures of location. There are 99 percentiles denoted P1, P2,... P99, which divide a set of data into 100 groups with about 1% of the values in each group. Converting from the kth Percentile to the Corresponding Data Value n total number of values in the data set k percentile being used L locator that gives the position of a value P k kth percentile Converting from the kth Percentile to the Corresponding Data Value Quartiles Are measures of location, denoted Q 1, Q 2, and Q 3, which divide a set of data into four groups with about 25% of the values in each group. Q 1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q 2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q 3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. 2

Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts Some Other Statistics Interquartile Range (or IQR): Q3 Q1 Semi interquartile Range: Midquartile: 10 90 Percentile Range: P 90 P 10 5

3 Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts Some Other Statistics Interquartile Range (or IQR): Q3 Q1 Semi interquartile Range: Midquartile: Percentile Range: P 90 P 10 5 Number Summary For a set of data, the 5 number summary consists of 1. the minimum value; 2. the first quartile Q1; 3. the median (or second quartile Q2); 4. the third quartile, Q3; and 5. the maximum value. Boxplot A boxplot (or box and whisker diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3. Boxplot of Movie Budget 3

Boxplots Normal Distribution Heights from a Simple Random Sample of Women Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches Outliers An outlier is a value that lies

4 Boxplots Normal Distribution Heights from a Simple Random Sample of Women Skewed Distribution: Salaries (in thousands of dollars) of NCAA Football Coaches Outliers An outlier is a value that lies very far away from the vast majority of the other values in a data set. Important Principles An outlier can have a dramatic effect on the mean. An outlier can have a dramatic effect on the standard deviation. An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured. Outliers for Modified Boxplots For purposes of constructing modified boxplots, we can consider outliers to be data values meeting specific criteria. In modified boxplots, a data value is an outlier if it is... Modified Boxplots Boxplots described earlier are called skeletal (or regular) boxplots. Some statistical packages provide modified boxplots which represent outliers as special points. above Q 3 by an amount greater than 1.5 IQR or below Q 1 by an amount greater than 1.5 IQR 4

5 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. Modified Boxplots Example Pulse rates of females listed in Data Set 1 in Appendix B. 3 4 Measures of Relative Standing and Boxplots In this section we have discussed: z Scores z Scores and unusual values Percentiles Quartiles Converting a percentile to corresponding data values Other statistics 5 number summary Boxplots and modified boxplots Effects of outliers 3 4 Measures of Relative Standing and Boxplots Always consider certain key factors: Context of the data Source of the data Sampling Method Measures of Center Measures of Variation Distribution Outliers Changing patterns over time Conclusions Practical Implications 5

132/3. Boxplots Shown below is a STATDISK generated boxplot of the durations (in hours) of flights of NASA s Space Shuttle. What do the values of 0, 166, 215, 269, and 423 tell us? TI: page 130 132/4.

6 132/3. Boxplots Shown below is a STATDISK generated boxplot of the durations (in hours) of flights of NASA s Space Shuttle. What do the values of 0, 166, 215, 269, and 423 tell us? TI: page /4. Boxplot Comparisons Refer to the two STATDISKgenerated boxplots shown below that are drawn on the same scale. One boxplot represents weights of randomly selected men and the other represents weights of randomly selected women. Which boxplot represents women? How do you know? Which boxplot depicts weights with more variation? TI: page /8. z Score for World s Tallest Man Bao Xishun is the world s tallest man with a height of in. ( or 7 ft, 8.95 in.). Men have heights with a mean of 69.6 in. and a standard devia tion of 2.8 in. a. What is the difference between Bao s height and the mean height of men? b. How many standard deviations is that ( the difference found in part ( a))? c. Convert Bao s height to a z score. d. Does Bao s height meet the criterion of being unusual by corresponding to a z score that does not fall between and 2? 133/10. z Scores for Heights of Women Soldiers The U. S. Army requires women s heights to be between 58 in. and 80 in. Women have heights with a mean of 63.6 in. and a standard deviation of 2.5 in. Find the z score corresponding to the minimum height requirement and find the z score corresponding to the maximum height requirement. Determine whether the minimum and maximum heights are unusual. 6

7 Percentiles. In Exercises 15 18, use the given sorted values, which are the numbers of points scored in the Super Bowl for a recent period of 24 years. Find the percentile corresponding to the given number of points / In Exercises 19 26, use the same list of 24 sorted values given for Exercises Find the indicated percentile or quartile /22. P / /26. P /28. Boxplot for Number of English Words A simple random sample of pages from Merriam Webster s Collegiate Dictionary, 11 th edition, was obtained. Listed below are the numbers of defined words on those pages, and they are arranged in order. Construct a boxplot and include the values of the 5 number summary /29. Boxplot for FICO Scores A simple random sample of FICO credit rating scores was obtained, and the sorted scores are listed below. Construct a boxplot and include the values of the 5 number summary. S32B

When comparing in different sets of, the deviations should be compared only if the two sets of data use the

When comparing in different sets of, the deviations should be compared only if the two sets of data use the CHEBYSHEV S THEOREM The (or fraction) of any data set lying within K standard deviations of the mean is always 2 the following statements: 1 1, K 1 K At least ¾ or 75% of all values lie within 2 standard