Mean, Median & Mode
Calculate: Mean
To calculate the mean:
- Sum (add) all of the scores in a variable
- Divide that sum by the number of scores in the distribution.
You probably already know this formula. A mean is the average of scores. We like to use it becuase we don’t have to arrange the scores in any particular order before calculating it. We just add up all the numbers and divide by the number of scores. In statistical vocab we “sum” the numbers and divided the sum by N (the number of scores).
∑X/N
Calculate the mean of these numbers:
7
6
5
5
5
4
3
The sum of the variable called X is 35. That is
∑X = 35
N (number of scores) is 7.
The mean of these scores is calculated by dividing 35 by 7. So, the mean of these scores is 5. That is:
= 5
Calculate: Median
Finding the median in a distribution of integers is relatively easy. When there is an odd number of scores: it is the one left over when counting in from either end. When there are an even number of scores, the median is whatever the middle two scores are (if they are the same) or the halfway point between the middle-most two scores when they differ from each other.
Medians are most often used when distributions are skewed. Indeed, when data is presented in medians, ask about the means. If they are quite different, the distribution is highly skewed, and the sample may not be as representative as you would like.
A median requires that we put the numbers in order (from high to low, or low to high). The median is the score in the middle (if there are an odd number of scores) or the average of the two middle-most scores (if there are an even number of scores). That too much work, so we prefer the mean.
There is no easy formula for median. To calculate the median, arrange the scores in order of magnitude from high to low or from low to high (it doesn’t matter which one you choose). Select the score in the middle.
Take these number, and arrangement from high to low:
2
9
4
7
8
Here they are arranged in a distribution:
9
8
7
4
2
Find the score in the middle. In the following numbers, the median is 7:
9
8
7
4
2
Calculate: Mode
The mode is the most popular score (most common). If you plot a distribution, the mode will be the highest spot on the distribution. It will be the top of the mountain. If your mountain has more than one peak, the distribution will be bimodal (2 high spots) or multimodal (several high spots).
There are two ways to calculate this popularity.
First, the mode may be found by sorting the scores and selecting the one most frequently given.
The mode of this distribution is 5:
11
9
5
5
5
2
Second, and more practical in a distribution of many scores, the mode is the highest point on a frequency distribution. If a frequency distribution is accurately drawn, both approaches will yield the same result.
In this case, there is one person who scored 2. Three who scored 5. One who scored, 9. And one who scored 11. So the highest point of this graph (histogram) is the mode.
When we make a distribution, the scores are arranged from left to right, with the lowest scores on the left and the highest scores on the right. When everyone has the same score, the distribution is a straight horizontal line. When more than one person has the same score, the scores are stacked vertically. Consequently, a distribution where everyone had the same score would be represented by a straight vertical line.
