Measure of Central Tendency - Mean, Median and Mode
A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary sstatistic:
1. Mean
2. Median
3. Mode
The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.
1. Mean (Arithmetic)
The mean (or average) is the most popular and well known measure of central tendency. It can be used with both discrete and continuous data, although its use is most often with continuous data . The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. So, if we have n values in a data set and they have values x1, x2, ..., xn, the sample mean, usually denoted by (pronounced x bar), is:
This formula is usually written in a slightly different manner using the Greek capitol letter, , pronounced "sigma", which means "sum of...":
Example:
The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below:
15 13 18 16 14 17 12
Find the mean of this set of data values.
Solution:
So, the mean mark is 15
2. Median
The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.
Example:
The marks of nine students in a geography test that had a maximum possible mark of 50 are given below:
47 35 37 32 38 39 36 34 35
Find the median of this set of data values.
Solution:
Arrange the data values in order from the lowest value to the highest value:
32 34 35 35 36 37 38 39 47
The fifth data value, 36, is the middle value in this arrangement.
2. Median
The median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value.
Example:
The marks of nine students in a geography test that had a maximum possible mark of 50 are given below:
47 35 37 32 38 39 36 34 35
Find the median of this set of data values.
Solution:
Arrange the data values in order from the lowest value to the highest value:
32 34 35 35 36 37 38 39 47
The fifth data value, 36, is the middle value in this arrangement.
In general:
If the number of values in the data set is even, then the median is the average of the two middle values.
3. Mode
The mode of a set of data values is the value(s) that occurs most often.
The mode has applications in printing. For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others.
Likewise, the mode has applications in manufacturing. For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others.
Example:
Find the mode of the following data set:
48 44 48 45 42 49 48
Solution:
The mode is 48 since it occurs most often.
Note:
• It is possible for a set of data values to have more than one mode.
• If there are two data values that occur most frequently, we say that the set of data values is bimodal.
• If there is no data value or data values that occur most frequently, we say that the set of data values has no mode.
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