Relative Standing and Boxplots Section 3-4 Relative Standing and Boxplots

Learning Targets This section introduces measures of relative standing, which are numbers showing the location of data values relative to the other values within a data set. They 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.

We will discuss the following topics: z-Scores Finding Percentile Finding Quartiles Box Plot Interquartile Range Modified Box Plots

z Score (or standardized value) Topic 1 - Z score z Score (or standardized value) the number of standard deviations that a given value x is above or below the mean

z-Scores What does it mean How do you find it The z-Score for a particular data point tells you the number of standard deviations the point is away from the mean 𝑧= 𝑥− 𝑥 𝑠

Population Sample x – µ x – x z = z = s  Measures of Position z Score Round z scores to 2 decimal places

Example 𝑧= 𝑥− 𝑥 𝑠 Consider that the mean height is 68.34 in with a standard deviation of 3.02 in. Also the mean weight is 172.55 lb with a standard deviation of 26.33 lb. A man is 76.2 in tall and 237.1 lb heavy. Which of these measurements is more extreme? 𝒛= 𝒙− 𝒙 𝒔 = 𝟕𝟔.𝟐−𝟔𝟖.𝟑𝟒 𝟑.𝟎𝟐 =𝟐.𝟔𝟎 𝒛= 𝒙− 𝒙 𝒔 = 𝟐𝟑𝟕.𝟏−𝟏𝟕𝟐.𝟓𝟓 𝟐𝟔.𝟑𝟑 =𝟐.𝟒𝟓 ROUND-OFF RULE: Round z-Scores to the nearest hundredth, as that is how they are typically plugged into statistical tables.

z Scores and Usual Values Whenever a data value is less than the mean, its corresponding z score is negative.

Interpreting Z Scores 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 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

Example: Helen Mirren was 61 when she earned her Oscar-winning Best Actress award. The Oscar-winning Best Actresses have a mean age of 35.8 years and a standard deviation of 11.3 years. What is the difference between Helen Mirren’s age and the mean age? b) How many standard deviations is that? c) Convert Helen Mirren’s age to a z score. d) If we consider “usual” ages to be those that convert to z scores between –2 and 2, is Helen Mirren’s age usual or unusual? 61 – 35.8 = 25.2 25.2 / 11.3 = 2.23 𝒛= 𝒙− 𝒙 𝒔 = 𝟔𝟏−𝟑𝟓.𝟖 𝟏𝟏.𝟑 =𝟐.𝟐𝟑 z = 2.23 > 2, her age is unusual.

Your Turn: Human body temperatures have a mean of 98 Your Turn: Human body temperatures have a mean of 98.60°F and a standard deviation of 0.72°F. Convert each given temperature to a z score and determine whether it is usual or unusual. a) 101.00°F b) 97.90°F c) 96.98°F 𝒂) 𝒛= 𝒙− 𝒙 𝒔 = 𝟏𝟎𝟏−𝟗𝟖.𝟔 𝟎.𝟕𝟐 =𝟐.𝟑𝟑𝟑>𝟐, 𝒖𝒏𝒖𝒔𝒖𝒂𝒍 𝒃) 𝒛= 𝒙− 𝒙 𝒔 = 𝟗𝟕.𝟗−𝟗𝟖.𝟔 𝟎.𝟕𝟐 =−𝟎.𝟗𝟕𝟐>−𝟐, 𝒖𝒔𝒖𝒂𝒍 𝒄) 𝒛= 𝒙− 𝒙 𝒔 = 𝟗𝟔.𝟗𝟖−𝟗𝟖.𝟔 𝟎.𝟕𝟐 =−𝟐.𝟐𝟓<−𝟐, 𝒖𝒏𝒖𝒔𝒖𝒂𝒍

Your Turn: Scores on the SAT test have a mean of 1518 and a standard deviation of 325. Scores on the ACT test have a mean of 21.1 and a standard deviation of 4.8. Which is relatively better: a score of 1840 on the SAT test or a score of 26.0 on the ACT test? Why? 𝑺𝑨𝑻: 𝒛= 𝒙− 𝒙 𝒔 = 𝟏𝟖𝟒𝟎−𝟏𝟓𝟏𝟖 𝟑𝟐𝟓 =𝟎.𝟗𝟗𝟏 𝑨𝑪𝑻: 𝒛= 𝒙− 𝒙 𝒔 = 𝟐𝟔−𝟐𝟏.𝟏 𝟒.𝟖 =𝟏.𝟎𝟐𝟏 𝑨𝑪𝑻 𝒔𝒄𝒐𝒓𝒆 𝒊𝒔 𝟏.𝟎𝟐𝟏 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 𝒂𝒃𝒐𝒗𝒆 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏 𝒉𝒐𝒘𝒆𝒗𝒆𝒓, 𝒕𝒉𝒆 𝑺𝑨𝑻 𝒔𝒄𝒐𝒓𝒆 𝒊𝒔 𝒐𝒏𝒍𝒚 𝟎.𝟗𝟗𝟏 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒅𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 𝒂𝒃𝒐𝒗𝒆 𝒕𝒉𝒆 𝒎𝒆𝒂𝒏. 𝑺𝒐 𝒕𝒉𝒆 𝑨𝑪𝑻 𝒔𝒄𝒐𝒓𝒆 𝒊𝒔 𝒓𝒆𝒍𝒂𝒕𝒊𝒗𝒆 𝒃𝒆𝒕𝒕𝒆𝒓.

ROUND-OFF RULE: Round off to the nearest whole number. Topic 2 - Percentile What does it mean How do you find it A percentile tells you what percentage of the data is less than a particular data value 𝑃 𝑥 = # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 <𝑥 𝑛 ⋅100 ROUND-OFF RULE: Round off to the nearest whole number.

𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟐𝟐 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟗𝟑 Example 𝑃 𝑥 = # 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 <𝑥 𝑛 ⋅100 The scores on the most recent quiz are posted in the table below. Assume you are the person that scored a 93, and calculate your percentile. 78 100 99 98 21 57 68 75 85 88 87 86 39 27 95 97 93 77 82 81 79 62 65 𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟐𝟐 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟗𝟑 𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒊𝒍𝒆= 𝟐𝟐 𝟑𝟎 𝟏𝟎𝟎  𝟕𝟑 𝒓𝒅

𝒂) 𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟗 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟒𝟕 𝒃) 𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟏𝟖 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟔𝟏 Your Turn: Use the given sorted values (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. 36 37 37 39 39 41 43 44 44 47 50 53 54 55 56 56 57 59 61 61 65 69 69 75 47 points b) 61 points 𝒂) 𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟗 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟒𝟕 𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒊𝒍𝒆= 𝟗 𝟐𝟒 𝟏𝟎𝟎  𝟑𝟖 𝒕𝒉 𝒃) 𝑻𝒉𝒆𝒓𝒆 𝒂𝒓𝒆 𝟏𝟖 𝒅𝒂𝒕𝒂 𝒗𝒂𝒍𝒖𝒆𝒔<𝟔𝟏 𝑷𝒆𝒓𝒄𝒆𝒏𝒕𝒊𝒍𝒆= 𝟏𝟖 𝟐𝟒 𝟏𝟎𝟎=𝟕𝟓 𝒕𝒉

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

Converting from the kth Percentile to the Corresponding Data Value

Example: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Find the indicated percentile. 36 37 37 39 39 41 43 44 44 47 50 53 54 55 56 56 57 59 61 61 65 69 69 75 a) P50 b) P22 𝑳= 𝒌 𝟏𝟎𝟎 ∙𝒏= 𝟓𝟎 𝟏𝟎𝟎 ∙𝟐𝟒=𝟏𝟐, data value 53 𝑳= 𝒌 𝟏𝟎𝟎 ∙𝒏= 𝟐𝟐 𝟏𝟎𝟎 ∙𝟐𝟒=𝟔, data value 41

The data must be ordered least to greatest first. Topic 3 - Quartiles What does it mean How do you find it Quartiles are denoted by 𝑄 1 , 𝑄 2 ,𝑎𝑛𝑑 𝑄 3 . They divide the data into 4 groups that each contain about 25% of the data. * The data must be ordered least to greatest first. ROUND-OFF RULE: Round L to the nearest whole number unless it is exactly at .5, then find the average of the #’s it is in between.

divide ranked scores into four equal parts Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts 25% Q3 Q2 Q1 (minimum) (maximum) (median)

Quartiles Are measures of location, denoted Q1, Q2, and Q3, which divide a set of data into four groups with about 25% of the values in each group. Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

5-Number Summary What does it mean Why do we do it? The 5-Number Summary of a data set is a table that gives the minimum value, 𝑄 1 , 𝑄 2 ,𝑎𝑛𝑑 𝑄 3 , and the maximum value The 5-Number Summary gives us all of the information we need in order to create a box plot (which we will learn next)

Example Find the 5-Number Summary for the following data set: 3 5 7 4 2 1 6 8 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒐𝒓 STAT  Edit, Enter data into L1 STAT  Calc, select “1-Var Stats”, List = L1 minX = 1, Q1=2.5, Med = 4.5, Q3 = 6.5, maxX = 8

Homework P.127-128: #7, 8, 15-18, #27(only complete 5 # summary)

Section 3.4 Day 2

Topic 4 - Box Plots How to construct it Why do we do it? On a number line, plot each of the values from your 5-Number Summary Place vertical lines at 𝑄 1 , 𝑄 2 , 𝑎𝑛𝑑 𝑄 3 Connect the tops and bottoms of 𝑄 1 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑄 3 Gives us an idea about the distribution, spread, and center of the data Great for comparing two sets of data

Boxplots Boxplot of Movie Budget Amounts

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

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

Example 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒐𝒓 STAT  Edit, Enter data into L1 Create a box plot using the following data about the number of times Abena solved a rubik’s cube in a single minute. 1 5 2 3 8 4 9 12 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒐𝒓 STAT  Edit, Enter data into L1 Press 2nd and “Y=” keys STAT PLOTS, select “Plot1” and turn to “On” Select the 5th icon in “Type” XList = L1 Quit Press “Zoom” and “9” keys minX = 1, Q1=2.5, Med = 4, Q3 = 8.5, maxX = 8

Your Turn: Use the given sorted values (the numbers of points scored in the Super Bowl for a recent period of 24 years). Construct a boxplot and include the values of the 5 – number summary. 37 37 39 39 41 43 44 44 47 50 53 54 55 56 56 57 59 61 61 65 69 69 75 minX = 36, Q1=42, Med = 53.5, Q3 = 60, maxX = 75 42 53.5 60 36 75

Topic 5 - Interquartile Range How to find it Why do we do it? 𝐼𝑄𝑅= 𝑄 3 − 𝑄 1 It helps us indicate any outliers. Anything greater than 𝑄 3 +1.5(𝐼𝑄𝑅) or less than 𝑄 1 −1.5(𝐼𝑄𝑅) is considered an outlier.

Topic 6 - Outliers and Modified Boxplots 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 . . . above Q3 by an amount greater than 1.5  IQR or below Q1 by an amount greater than 1.5  IQR

Modified Boxplots Boxplots described earlier are called skeletal (or regular) boxplots. Some statistical packages provide modified boxplots which represent outliers as special points. 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒐𝒓 STAT  Edit, Enter data into L1 Press 2nd and “Y=” keys STAT PLOTS, select “Plot1” and turn to “On” Select the 4th icon in “Type” XList = L1 Quit Press “Zoom” and “9” keys

Modified Box Plot What is it? Why do we do it? A modified box plot follows the same procedure as a normal box plot, except you distinguish outliers using asterisks and stop your line at the least and greatest values that aren’t outliers. Outliers can significantly effect the shape of the data, so using the modified box plot makes are representation resistant.

Example Create a modified box plot using the following data about the number of times Ms. P served an ace against Mitch after school at the tennis courts. ACES 32 35 37 28 30 42 45 41 49 29 120 30 37 45 28 49 120  Modified Boxplot 30 37 45 28 120 Regular Boxplot

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.

Recap 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 Outliers and Boxplots and modified boxplots Effects of outliers

Putting It All Together 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

Homework P.126-128: #4, 11, 14, 27, 28