CHAPTER 3

3.1 MEASURES OF CENTRAL TENDENCY FOR UNGROUPED DATA3.2 MEAN, VARIANCE, AND STANDARD DEVIATION FOR GROUPED DATA

MEASURES OF CENTRAL TENDENCY

In chapter 2 we discussed how to summarize data using different methods and to display data using graphs. Graphs are one important component of statistics; however it is also important to numerically describe the main characteristics of a data set. The numerical summary measures, such as the one to identify the center and spread of distribution, identify many important features of a distribution.

3.1 MEASURES OF CENTRAL TENDENCY FOR UNGROUPED DATA

A measure of central tendency gives the center of a histogram or a frequency distribution curve.

I) Mean

Obtained by dividing the sum of all values by the number of values in the data set. Thus,

Mean for population data: μ=∑ xN

Mean for sample data: x =

∑ xn

Example:

The following are the ages of all eight employees of a small company:

53 32 61 27 39 44 49 57

Find the mean age of these employees

Example:

The following are the ages of 20 persons empanelled for jury duty by a court:

48 58 33 42 57 61 4931 52 25 46 60 53 3052 63 41 34 47 38

Find their mean age.

II) Median

The value of the middle term in a data set that has been ranked in increasing order:

-rank the data set in increasing order. -Find the middle term. The value of this term is the median.

Example:

Find the median.

10 5 19 8 3

Example:

Ten persons sent out to interviews 50 students at each of 10 different campuses, found that 18, 13, 15, 12, 8, 21, 7, 11, 16 and 3 of the students sampled jog regularly. Find the median.

I) Mode

The value that occurs with the highest frequency in a data set.

Example:

Determine the mode for this set of data.

4 4 5 2 4 3 3 3 33 4 5 5 4 5 6 5 3

Example:

Find the mode.

3.2 MEAN, MEDIAN, MODE, VARIANCE, AND STANDARD DEVIATION FOR GROUPED DATA

I) Mean for Grouped Data

14

12 20 18 21 15 9 16 17 18 18

9 14 11 13 11 12 15 20 14 15 1110

8 20 15 13 9 19 10 14 26 15

14

8 12 19 12 10 13 15 9 15 24

12

13 14 12 15 7 16

Mean for population data: μ=∑ mfN

Mean for sample data: x =

∑ mfn

m =midpointf = frequency of class

Example:

Calculate the mean of daily commuting times.

Daily Commuting Number of Employees1 to 10 411 to 20 921 to 30 631 to 40 441 to 50 2

II) Median

Median for Grouped Data

m = b + i [n2−lfm ]

b = lower bound for median classi = class intervaln = number of observationl = cumulative frequency before median classfm = frequency for median class

Example:

Find median based on data below:

II) Mode

For grouped data, the mode can be approximated by the “midpoint of the class containing the largest number of class frequency”

Class Frequency, f10-14 615-19 1920-24 2325-29 1830-34 935-39 340-44 2

Mode = L + [ f o−f 1( f o−f 1)+( f o−f 2) ] x c

L = lower class boundary of the class containing modeC = size of the class containing modefo = frequency of the class containing modef1 = frequency of the class before class containing modef2 = frequency of the class after the class containing mode

Example:

Find mode for grouped data below.

Class Interval Number of employee1-4 165-8 209-12 28

13-16 2417-20 1621-24 1125-28 5

Tutorial 3

1. The following data give the numbers of car thefts that occurred in a city during the past 12 days.

6 3 7 11 4 3 8 7 2 6 9 15

Find the mean, median and mode for the above data.

2. A spot check of the number of students attending a lecture over a 21-day period was made, with the following results:

78 87 94 92 79 78 6782 88 62 97 80 82 6290 79 82 90 87 86 62

Determine the mean, median and mode.

3. The following table gives the frequency distribution of the number of hours spent per week playing video games by all 60 students of the eighth grade at a school.

Find the mode, median and mean for the above data.

4. For 50 airplanes that arrived late at an airport during a week, the time which they were late was observed. In the following table, x denotes the time (in minutes) by which an airplane was late and f denotes the number of airplanes.

Find the mode, median and mean for the above data.

Hours per week Number of students1– 5 76-10 12

11-15 1516-20 1321-25 826-30 5

x F1 – 20 14

21 – 40 1841 – 60 961 – 80 581 – 100 4