DATA ANALYSIS

DATA ANALYSIS

Measures of Central Tendency

MEAN

MODE

MEDIAN

Measures of Central Tendency

MEAN

n

xx

scores of Number

Scores the of Sum

MEDIAN is the middle score when the scores are arranged in numerical order

MODE is the most frequently occurring score in the data

[It is possible to have no mode or more than one mode]

Example 1- Find the mean, mode and median of the following set of scores

3, 5, 2, 7, 8, 8, 9, 10

Mean = 8109887253

= 6.5

852

2, 3, 5, 7, 8, 8, 9, 10

5.7287

Note that when you have an even number of scores the median is the mean of the two middle scores

Arrange the scores in order

Mode = 8

Example 1 Continued

Median =

Example 2 - Find the mean, mode and median of the following set of scores.

67, 88, 43, 76, 75, 82, 71

Mean = 7

71827576438867

= 71.7143

7502

43, 67, 71, 75, 76, 82, 88

Median = 75

All scores occur the same number of times, so there is no mode in this case

Arrange the scores in order

No Mode

Example 2 Continued

Example 3 - Find the mean, mode and median of the following set of scores.

12, 18, 23, 16, 15, 12, 11, 16, 20, 14, 12, 22, 16, 14, 22

Mean = 15

221416221214201611121516231812

= 16.2

15243

11, 12, 12, 12, 14, 14, 15, 16, 16, 16, 18, 20, 22, 22, 23

Median = 16

When we have two modes, the data is bi-modal

Arrange the scores in order

Mode = 12 and 16

Example 3 Continued

MEASURES OF SPREAD

Range

Interquartile Range

Standard Deviation

MEASURES OF SPREAD

Range = Highest score – Lowest score

Interquartile Range

= Upper Quartile – Lowest Quartile

The upper quartile is the median of the top 50% of the scores whilst the lower quartile is the median of the bottom half of the scores.

The interquartile range looks at the middle 50% of the scores and measures the range in this set.

Standard Deviation

n

xxs

2

The advantage of using standard deviation as a measure of spread is that it uses all scores.

22

xnxs or

The standard deviation measures the deviations from the mean.

Example 1- Find the range, interquartile range and standard deviation of the following set of scores.

3, 5, 2, 7, 8, 8, 9, 10

Range = 10 – 2

= 8Remember to subtract

Range = Highest score – Lowest score

Example 1 continued - Finding the interquartile range

Divide the scores into two sets

Interquartile Range = 8.5 – 4

Arrange the scores in order

2, 3, 5, 7, 8, 8, 9, 10

= 4.5

Find the middle score of each half

Example 1 continued - Finding the standard deviation

Using subtract

6.5 from each score

Mean = 6.5 2, 3, 5, 7, 8, 8, 9, 10

Square each of these values

n

xxs

2

-4.5, -3.5, -1.5, 0.5, 1.5, 1.5, 2.5, 3.5

20.25, 12.25, 2.25, 0.25, 2.25, 2.25, 6.25, 12.25

n

xxs

2

858 69.225.7

That method is too complicated and hence is very rarely used.

Let’s consider the second formula for calculating standard deviation.

22

xnxs

Example 1 continued

Using square

each score

Mean = 6.5 2, 3, 5, 7, 8, 8, 9, 10

4, 9, 25, 49, 64, 64, 81, 100

25.68396

25.425.49

22

xnxs

22

xnxs

69.2

Generally we do not calculate the standard deviation in this manner. We use our calculator in statistics mode.

The symbol used to represent standard deviation varies from calculator to calculator.

nxSome examples are and

Example 2- Find the range, interquartile range and standard deviation of the following set of scores.

67, 88, 43, 76, 75, 82, 71

Range = 88 – 43

= 45Remember to subtract

Range = Highest score – Lowest score

Example 2 continued

Divide the scores into two sets

Interquartile Range = 82 – 67

Arrange the scores in order

43, 67, 71, 75, 76, 82, 88

= 15

Find the middle score of each half

Example 2 continued

Mean = 71.71

1849, 4489, 5041, 5625, 5776, 6724, 7744

271.717

37248

94.514214.5321

Using square

each score2

2

xnxs

22

xnxs

4.13

43, 67, 71, 75, 76, 82, 88

Frequency TablesFrequency tables are a good way to present data.

The first column is the score, the second column shows the frequency or number of times the score in the first column occurred.

Standard Deviation = 2

2

xf

fx

Formulas used with frequency tables.

Mean = f

fx

Frequency Tables

Mean = f

fxfx

45801191629520

521

30521

37.17

Score Frequency

15 316 517 718 919 520 1

Total 30

Example - Calculate the mean, mode, range and standard deviation for the data in the table.

Frequency Tables - Finding the standard deviation

fx

45801191629520

521

Score Frequency

15 316 517 718 919 520 1

Total 30

675

2fx

1280202329161805400

9099

Standard Deviation

22

xf

fx

237.17

309099

3.1