Notes

Statistics/Part 2:

Measures of Central Tendency
Researchers use statistics to represent and summarize the data they collect. Large-scale studies involve gathering and analyzing huge amounts of data. Thus, researchers use a score or numerical value to represent the information gathered from a sample. A central tendency is a value that represents the distribution of responses in a sample. The three main measures of central tendency are mean, mode, and median.

Mean: The mean is the most widely used measure of central tendency and the easiest one to calculate. This measure of a set of scores is the average of the scores; that is, it equals the total of the scores (values) divided by the number of scores (number of people). Consider a sample consisting of 10 individuals who have taken an IQ test. If their IQ scores are 105, 115, 120, 110, 135, 125, 100, 130, 105, and 105, we would calculate the mean value of this information as follows:

1,150 (total of the IQ scores) ÷ 10 (number of people taking the test) = 115
Median: Another measure of central tendency, the median is the value or score in a given set of scores (arranged in increasing or decreasing order) that divides the set into two halves. Let’s consider our previous example and list the IQ scores in ascending order: 100, 105, 105, 105, 110, 115, 120, 125, 130, and 135. The values that divide this information into two halves are 110 and 115, i.e., there are exactly four scores below and above these two values. In this case, the median will be the average of the two scores:

(110 + 115) ÷ 2 = 112.5

Mode: The third measure of central tendency is the mode. The mode refers to the score that appears the most number of times in a given set of information. The most frequent score in the set of scores is thus the modal value of the data. As you can see, the most frequently recurring score in our previous example (i.e., the mode) is 105.

The most popular measure of central tendency is the mean, followed by the median, and then the mode. The mode will not apply to every set of information because the data might only have scores that appear just once and none that repeat.
Measures of Dispersion
Dispersion is a statistical value that gives an idea of the spread or variation among the scores in a set. This measure helps researchers determine the extent to which the values in a data set vary. Two common measures of dispersion are range and standard deviation.

Range: The range of a data set is the difference between its highest and lowest values or scores. In our example of the IQ test, the highest score was 135 and the lowest score was 100. The difference between these two values, 35, is the range of the data set. For data sets with values that show a high amount of variation, the range is usually not an accurate measure of dispersion.
Standard deviation: Standard deviation is the most common measure of dispersion used in psychological studies involving different groups or sample sets from a given population. It shows the extent to which each individual score in a given set varies from the mean of that set. Thus, standard deviation is a measure of dispersion only when a researcher uses the mean as a measure of central tendency. This measure of dispersion is crucial because different sets of data might have identical means, but contain huge differences in terms of variability. The lower the standard deviation is, the more reliable the data. The formula for calculating standard deviation (SD) is given as:

In this formula, represents each individual score. is the mean value, and is the number of scores. Let’s return to our previous example of IQ test scores. In the table, the last column shows the sum of the squared values of the differences between the mean value and each of the individual scores, which is 1,300. Let’s insert the values in the formula:

standard deviation =(x-x_)2

The standard deviation for the given data set is 12.01, which means the individual scores differ from the mean by an average value of 12.01.

Median is a value in a data set that divides the set in two halves. Mean is the average of all of the values in a data set. Mode is a value that appears most frequently in a data set.