DATA ANALYSIS

n Measures of Central Tendency F MEAN F MODE F MEDIAN

n Measures of Central Tendency MEAN

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 = = 6.5

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

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 = = 16.2

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 n Range n Interquartile Range n Standard Deviation

MEASURES OF SPREAD n Range = Highest score – Lowest score n 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 The advantage of using standard deviation as a measure of spread is that it uses all scores. 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 = 8 Remember 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 -4.5, -3.5, -1.5, 0.5, 1.5, 1.5, 2.5, , 12.25, 2.25, 0.25, 2.25, 2.25, 6.25, 12.25

That method is too complicated and hence is very rarely used. Let’s consider the second formula for calculating standard deviation.

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

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. Some 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 = 45 Remember 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 = , 4489, 5041, 5625, 5776, 6724, 7744 Using square each score 43, 67, 71, 75, 76, 82, 88

Frequency Tables Frequency 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 = Formulas used with frequency tables. Mean =

Frequency Tables Mean = fx Example - Calculate the mean, mode, range and standard deviation for the data in the table.

Frequency Tables - Finding the standard deviation fx Standard Deviation