The mean is the most common measure of central tendency, and it will be an important concept throughout the course. Computing the mean requires scores that are numerical values measured on an interval or ratio scale. The mean is calculated by taking the sum of all the scores, and then by dividing this sum by the number of scores in the in set.
For sample data the mean is: M = ΣX/n (n being the sample size)
For population data the mean is:
μ = ΣX/N (N being the size of the population)
Conceptually, you can think of the mean as the amount that every individual would get if the total (ΣX) is divided equally between all the individuals (n or N, depending on whether it is a sample of people or the whole population). The mean is the balance point of the distribution, with equal ‘weight’ to each side of it.
Because the mean is based on every score in the set of scores, changing a score will also change the mean. Similarly, adding or removing scores will also almost always change the mean, unless the score that is added or remove is exactly equal to the mean.
Although the mean is the most commonly used measure of central tendency, there are situations in which the mean does not provide an accurate reflection of the distribution as a whole. We will come back to this point after introducing the other two measures of central tendency.
A second measure of central tendency is the median. The median is the value that corresponds to the exact midpoint of all the values. Thus, if you have a list of scores and order them going from lowest to highest, the median would be the score that is the midpoint of this list. The median divides the set of scores in such a way that 50% of the scores is smaller or equal to the median, and 50% is greater or equal to the median. Computing the median requires the scores to be put in rank order, and therefore the scores must come from an interval, ordinal, or ratio scale. Usually, the median can be found by the following counting procedure:
- With an odd number of scores, list the scores in order. The median is the middle value in the list of scores.
- With an even number of scores, list the scores in order. The median is the halfway between the two middle values in the list of scores. To get the halfway point, sum the two scores in the middle and divide the sum by 2.
A third measure of central tendency is the mode. The mode is defined as the most frequently occurring score in the distribution of scores. In a frequency distribution graph, the mode corresponds to the highest point on the graph. The mode can be determined for any scale, be it nominal, ordinal, interval, or ratio. The primary value of the mode is that it is the only measure of central tendency that can be used for measurements on a nominal scale.
Unlike the other two measures, a distribution can have more than one mode. If there is only one prominent peak in the distribution, the shape of the distribution is unimodal. If there are two prominent peaks, the distribution is called bimodal.
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The arithmetic mean is the most common measure of central tendency. It is computed by summing all the scores (sigma or Σ) and dividing by the number of scores (N):
Where X is the mean, ∑x is the addition or summation of all scores, and N is the number of cases.
- Example of calculating mean with formula:
Given the scores of first year students in a Statistics test, calculate the mean.
10 5 9 8 6 5 9 8 7 6 5 6
1. To calculate the mean, first add all scores; that is, 10+5+9+8++6+5+9+8+7+6+5+6= 84
2. Then divide the result by the number of cases (the number of scores): 12
3.
Applying the formula:
X= 84/12= 7
Thus, the mean or average score of this Statistics test is 7.
- Example of calculating the mean using a frequency table.
In this example, you are given a table of frequencies of the scores obtained in a Statistics test. The column on the left gives you test scores, and the column on the right the frequency (how many students obtained that score).
| X (score) | Frequency |
| 10 | 1 |
| 5 | 3 |
| 8 | 2 |
| 2 | 5 |
| 4 | 5 |
1. First, multiply each score by its frequency to calculate the sum of all scores:
10X1+5x3+8x2+2x5+4X5= 71
2. Then divide by the number of scores, which is the sum of all the frequencies: 1+3+2+5+5= 16
3. Applying the formula: 71/16= 4.43
The mean is sensitive to outliers (that is, unusually large or small observations). A 5% trimmed mean is calculated when there are outliers in the distribution, as it calculates the mean of the distribution when the top and bottom 5% scores are removed.
In this example, the 5% trimmed mean and the arithmetic mean are very similar. Thus, there are no extreme scores or outliers in this distribution that may be affecting the mean.
In this example, the 5% trimmed mean is different from the arithmetic mean. This implies that there are outliers in the distribution.