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Objective 3-3 Today we learn mean, median, and mode for grouped and ungrouped frequency distributions. Summarize data using the measures of central tendency, such as the mean, median, mode, and midrange. 3

3-2 The Sample Mean for an Ungrouped Frequency Distribution 3-13 13

3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-14 Score, X Frequency, f Score, X Frequency, f 2 2 1 4 1 4 2 12 2 12 3 4 3 4 4 3 4 3 5 5 14

3-2 The Sample Mean for an Ungrouped Frequency Distribution - Example 3-15 1) Create a new column (f times x) Score, X Frequency, f f × X 2 1 4 12 24 3 Score, X Frequency, f × f X 2 1 4 4 2 12 24 2) Find the sum of your new column (Σf∙X = 52) 3 4 12 4 3 12 5 5 3) Use the mean formula 15

3-2 The Sample Mean for a Grouped Frequency Distribution 3-16 The mean for a grouped frequency distributu ion is given by X f n Here the correspond ing class midpoint. m = ( ) × å . 16

3-2 The Sample Mean for a Grouped Frequency Distribution - Example 3-17 Create a new column for the midpoint of each group Class Frequency, f Class Frequency, f 15.5 - 20.5 3 15.5 - 20.5 3 20.5 - 25.5 5 20.5 - 25.5 5 25.5 - 30.5 4 25.5 - 30.5 4 30.5 - 35.5 3 30.5 - 35.5 3 35.5 - 40.5 2 35.5 - 40.5 2 5 5 17

3-2 The Sample Mean for a Grouped Frequency Distribution - Example 3-18 Midpoint of each class Frequency times Midpoint Class Frequency, f X m f × 15.5 - 20.5 3 18 54 20.5 - 25.5 5 23 115 25.5 - 30.5 4 28 112 30.5 - 35.5 33 99 Class Frequency, f X × f X m m 15.5 - 20.5 3 18 54 5 23 115 25.5 - 30.5 4 28 112 30.5 - 35.5 3 33 99 35.5 - 40.5 2 38 76 20.5 - 25.5 Sum of this column is 17 (n = 17) Sum of this column is 456 35.5 - 40.5 2 38 76 5 5 18

3-2 The Sample Mean for a Grouped Frequency Distribution - Example 3-19 19

3-2 The Median-Ungrouped Frequency Distribution 3-28 For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value. 28

3-2 The Median-Ungrouped Frequency Distribution 3-29 If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above. In other words, add up your frequencies and divide by 2 29

3-2 The Median-Ungrouped Frequency Distribution - Example 3-30 LRJ Appliance recorded the number of VCRs sold per week over a one-year period. The data is given below. No. Sets Sold Frequency 1 4 2 9 3 6 5 30

3-2 The Median-Ungrouped Frequency Distribution - Example 3-31 No. Sets Sold Frequency 1 4 2 9 3 6 5 To locate the middle point, divide n by 2; 24/2 = 12. Locate the point where 12 values would fall below and 12 values will fall above. Consider the cumulative distribution. The 12th and 13th values fall in class 2. Hence MD = 2. n = 24 31

3-2 The Median-Ungrouped Frequency Distribution - Example 3-32 This class contains the 5th through the 13th values. 32

3-2 The Median for a Grouped Frequency Distribution 3-33 class median the of boundary lower L width w frequency f preceding immediately cumulative cf frequencies sum n Where MD can be computed from: The m = ) ( 2 + - Keep this written somewhere handy 33

3-2 The Median for a Grouped Frequency Distribution - Example 3-34 Class Frequency, f Class Frequency, f 15.5 - 20.5 3 15.5 - 20.5 3 20.5 - 25.5 5 20.5 - 25.5 5 25.5 - 30.5 4 25.5 - 30.5 4 30.5 - 35.5 3 30.5 - 35.5 3 35.5 - 40.5 2 35.5 - 40.5 2 5 5 34

3-2 The Median for a Grouped Frequency Distribution - Example 3-35 f Class Frequency, f Cumulative Frequency 15.5 - 20.5 3 20.5 - 25.5 5 8 25.5 - 30.5 4 12 30.5 - 35.5 15 35.5 - 40.5 2 17 Class Frequency, f Cumulative cf Median Class Frequency 15.5 - 20.5 3 3 Lm 20.5 - 25.5 5 8 25.5 - 30.5 4 12 w = 30.5-25.5 = 5 30.5 - 35.5 3 15 35.5 - 40.5 2 17 n=17 so MD is 17/2 or 8.5 5 5 35

3-2 The Median for a Grouped Frequency Distribution 3-37 n = 17 cf = 8 f = 4 w = 25.5 – 20.5 = 5 L = 25 . 5 m ( n 2 ) - cf (17 / 2) – 8 MD = ( w ) + L = ( 5 ) + 25 . 5 f m 4 = 26.125. 37

3-2 The Mode for an Ungrouped Frequency Distribution - Example 3-42 Values Frequency, f Values Frequency, f 15 3 Mode 15 3 20 5 20 5 25 8 25 8 30 3 30 So the mode is half of that, or 10.5 Σf = 21 3 35 2 35 2 5 5 42

3-2 The Mode for a Grouped Frequency Distribution - Example 3-44 Modal Class So the mode is half of that, or 10 Σf = 20 44