Measure of Central Tendency
Measure of Central Tendency forms an important part of Statistics more commonly known as Average. But average and Central Tendency are not one and the same things. Average, we know from the childhood days when we used to find out our average marks. The method of computing an average is fairly simple.
Average, in the statistical language is called the Arithmetic mean, or simply Mean.
Consider we have four test results, 90, 84, 76, 95
The mean of these four units would be the sum total of all, divided by the number of items, in this case, four. So the average for the give example will be (90+84+76+95)/4 which is 86.25.
The total numbers of items are often denoted by ‘n’, and the items of interest, i.e. the variables are called frequency. For the ease of notation, variables are denoted as ‘a’. This gives us the formula for computing mean which can be denoted as the sum total of, a1,a2, a3 and so on, divided by n where n represents the number of items (1, 2, 3 and so on)
Arithmetic Mean = (a1 a2 a3 a4)/n
Further Examples of Arithmetic Mean.
In a class heights of boys were calculated and were found as follows: 162, 150, 155, 160, 150, 169, 170, 160, 155, 150, 164, 167, 162 and 158 160.
The first step is to create a table with the number of heights in one column and the number of times they appear in another.
The mean of the following data will be computed after multiplying the number of frequencies to their corresponding heights and then dividing the sum by the total number of frequencies.
So the mean height will be 159.46
The next average in statistics is the Median
Median is the central value of the data when arranged in the ascending order. In statistics of higher level, median is used as a measure of dispersion. The median is the number which is positioned exactly at the middle of the series. In a series of odd number of values, the centre value is the median, in a series with even number of units, the sum of the two middle number divided by two is the median.
The formula for computing media in an odd series is [(number of units) + 1]/2
Example: compute the median of the following data.
8, 4, 7, 5, 10, 9, 1, 4, 2
The first task is to arrange the data in an ascending order
1, 2, 4, 4, 5, 7, 8, 9, 10
The median is the central value, apart from using the formula, striking out one value from the right and one from the left until we’re left with one will also give us the median.
1, 2, 4, 4, 5, 7, 8, 9, 10
So from this we can see that the median of the following series is 5.
However, in a series with even number of items, we will have two central values; the median is obtained there by finding the average of those two central values. This can be explained by the help of this illustration.
4, 5, 3, 6, 7, 3, 2, 8
The data will be arranged in an ascending order
2, 3, 3, 4, 5, 6, 7, 8
The median of the following data will be the average of 4 and 5 which is, (4+5)/2=4.5
The next measure of central tendency is the Mode. Mode, simply put is the value which has the highest number of recurrence in the series. In a frequency distribution table, mode is the value where the series reaches its peak, if plotted on a histogram.
The method of finding modal value in a quantitative distribution is fairly simple.
Example: Find mode in the following distribution 15, 6, 2, 3, 11, 3, 5, 7, 9, 3
The mode of the following distribution will be three which repeats itself thrice, more than any another unit in the distribution.
A group can sometimes have more than one modal value then the set is called Bimodal or Multimodal. If all the variables in a group have the same frequency, none of them repeating, or each repeating once, twice, and so on, then the group is said to have no modal value. If the two adjacent values are same, the average of the two values is the mode. In a frequency distribution table, the set containing the highest frequency is the modal frequency; this can be further elaborated with the help of an example:
If the question is, find the modal value, the normal answer would be 21 in the first instance, which is wrong. The modal value in this series is the frequency that corresponds to the highest number of frequency, thus the modal interval will be 250-300 which contains the highest number of values.
Range, is simply the range of data between the highest and the lowest value. The computation of range is a fairly simple process. The difference between highest and the lowest value in a series is called the Range.
Consider the following set of data:
40, 45, 43, 47, 38, 35, 50, 49, 37
The highest value of the series is, 50 and lowest value of the series is 37 therefore the range of the following series will be 50-37 which is 13.
The measure of central tendency attempts to find out one figure which represents the whole set of data. There are different methods and different application of using each which depend upon case to case basis. There are many measure of central tendency but these four are the most common ones.