Measures of Central Tendency (0-12) Objective: Calculate measures of central tendency, variation, and position of a set of data.

Measures of Central Tendency Measures of central tendency or center are numbers used to represent a set of data. Three types of measures of central tendency are mean, median, and mode.

Measures of Central Tendency The mean of a set of data is also known as the average. –The mean is the sum of the numbers in a set of data divided by the number of items. –The symbol for mean is x.

Measures of Central Tendency The median is the middle number in a set of data when the data are arranged in numerical order. –If there is an odd number of data values, the median is the middle number. –If there is an even number of data values, the median is the mean of the two middle numbers. –The abbreviation for median is med.

Measures of Central Tendency The mode is the number or numbers that appear most often in a set of data. –If there is a tie between numbers occurring the most, all numbers in the tie are the mode. –If no item appears most often, the set has no mode.

Measures of Variation Measures of variation or spread are used to describe the distribution of the data. One measure, the difference between the greatest and the least data values, is called the range.

Example 1 The table shows the number of laps Erika swam each day. Find the mean, median, mode, and range. DayNumber of Laps Saturday6 Sunday8 Monday8 Tuesday6 Wednesday4 Thursday9 Friday5 4, 5, 6, 6, 8, 8, 9 Mean = 46 / 7 = 6.6 Median = 6 Mode = 6 and 8 Range = 9 – 4= 5

Check Your Progress Choose the best answer for the following. A.Find the mean of the following: 5, 8, 4, 6, 2, 11. A.5.5 B.6 C.9 D / 6

Check Your Progress Choose the best answer for the following. B.Find the median of the following: 5, 8, 4, 6, 2, 11. A.5 B.5.5 C.6 D.9 2, 4, 5, 6, 8, / 2

Check Your Progress Choose the best answer for the following. C.Find the mode of the following: 5, 8, 4, 6, 2, 11. A.2 B.5 C.11 D.No Mode 2, 4, 5, 6, 8, 11

Check Your Progress Choose the best answer for the following. D.Find the range of the following: 5, 8, 4, 6, 2, 11. A.2 B.6 C.9 D – 2

Quartiles In a set of data, the quartiles are the values that separate the data into four equal subsets, each containing one fourth of the data. Q 1, Q 2, and Q 3 are used to represent the three quartiles. Q 1 is the lower quartile. It divides the lower half of the data into two equal parts. Q 2 is the median since it separates the data into two equal parts. Q 3 is the upper quartile. It divides the upper half of the data into two equal parts.

Example 2 Find the lower quartile, median, and upper quartile of the data shown below. 27, 25, 44, 13, 29, 44, 52, 28, 41 13, 25, 27, 28, 29, 41, 44, 44, 52 Q 1 = 52 / 2 = 26 Q 2 = 29 Q 3 = 88 / 2 = 44

Check Your Progress Choose the best answer for the following. –Calculate the quartiles for the following data set: 11, 19, 5, 34, 9, 25, 28, 16, 17, 11, 12, 7. A.Q 1 = 19.5, Q 2 = 26.5, Q 3 = 14 B.Q 1 = 10, Q 2 = 14, Q 3 = 22 C.Q 1 = 9, Q 2 = 12, Q 3 = 19 D.Q 1 = 11, Q 2 = 16, Q 3 = 25 5, 7, 9, 11, 11, 12, 16, 17, 19, 25, 28, 34

Box-and-Whisker Plots Data can be organized and displayed by dividing a set of data into four parts using the median and quartiles. This is a box-and-whisker plot. The box in a box-and-whisker plot represents the interquartile range. The interquartile range (IQR) is the difference between the upper and lower quartiles. Data that are more than 1.5 times the value of the interquartile range beyond the quartiles are called outliers.

Box-and-Whisker Plots To draw a box-and-whisker plot, use the following steps: –Draw a number line that includes the least and greatest numbers in the data. –Place dots above the number line to represent the three quartile points, any outliers, the least number that is not an outlier, and the greatest number that is not an outlier. –Connect the minimum number that is not an outlier to the lower quartile. –Connect the maximum number that is not an outlier to the upper quartile. –Draw a rectangle with one side at the lower quartile and the other side at the upper quartile. –Draw a vertical line inside the box through the median point.

Example 3 Mrs. Heflin is comparing the performances of her class on the past two chapter tests. a.Draw a double box-and- whisker plot for the data. Test Scores Chapter 5Chapter , 76, 77, 79, 79, 79, 81, 81, 83, 83, 84, 85, 86, 88 IQR = 84 – 79 = 5 79 – 1.5(5) = (5) = 91.5 Since there are no data points below 71.5 or above 91.5, there are no outliers.

Example 3 Chapter 5

Example 3 Mrs. Heflin is comparing the performances of her class on the past two chapter tests. a.Draw a double box-and- whisker plot for the data. Test Scores Chapter 5Chapter , 65, 68, 69, 70, 72, 77, 86, 88, 88, 91, 91, 93, 94 IQR = 91 – 69 = – 1.5(22) = (22) = 124 Since there are no data points below 36 or above 124, there are no outliers.

Example 3 Chapter 5 Chapter 6

Example 3 b.Use the box-and-whisker plot to compare the data.  The medians were about the same, but the IQRs were very different.  The spread of the scores was much greater for Chapter 6. Chapter 5 Chapter 6