Central tendency and standard deviation

Central tendency and standard deviation

The most common types of central tendency:

• Mode: The most frequently occurring number in a list. E.g. (1, 2, 2, 3, 3, 3, 4) has a mode of 3. If several members of the list occur equally often the list has several modes.
• Median: The median is the middle number of the group when they are ranked in order. If there is an even number of numbers, the mean of the middle two is taken. E. g. the above list has a median of 3. The list (1, 3, 7, 13) has a median of 5.
• Arithmetic mean: The arithmetic mean is the sum of the values in the list divided by their number. E.g. the arithmetic mean of the list (1, 2, 2, 2, 3, 3, 3, 4, 5, 5) is (1 + 2 + 2 + 2 + 3 + 3 + 3 + 4 + 5 + 5) / 10 = 30 / 10 = 3.

Range and Interquartile Range (IQR)

The range is calculated by subtracting the smallest observation (sample minimum) from the greatest (sample maximum) and provides an indication of statistical dispersion. For the number of correct items in the example table (Table 1), the range is 20 - 2 = 18.

The IQR is also called the midspread or middle fifty. It’s a measure of statistical dispersion, being equal to the difference between the third and first quartiles: IQR = Q3 - Q1. For the number of correct items in Table 1 the IQR is 14.75 - 10.50 = 4.25. The IQR is important for boxplots.


Figure 2: A boxplot of the token numbers of correct items

Standard deviation

The standard deviation (SD) is a measure of the average deviation from the arithmetic mean.

• The SD tells us how the results cluster around the arithmetic mean. A low SD means that all results are relatively close to the mean, a large SD indicates that the results have a higher dispersion.

• The SD is calculated as follows:
  ,
  with s indicating the standard deviation, N the total number of results, and x each deviation from the arithmetic mean.

• Excel can calculate the SD for you (function: STABW.S).

Exercise 3: Calculate (with a simple calculator) the SD of the values 2, 3, 4 and 5.

Exercise 4 : Calculate (in Excel) the mean, the range and the standard deviation for the percentage of correct items in example table 1.