Measures of Central Tendency
If we take the achievement scores of the students of a class, we see that the marks of most of the students lie somewhere between the highest and the lowest scores of the whole class. This tendency of a group of distribution is named as the central tendency.
According to -
Tate - Central Tendency is a sort of average or typical value of the items in the series and its function is to summarize the series in trams of this average value.
The most common measures of central tendency are -
Arithmetic mean or mean
Median
Mode
The values of mean, median or mode also helps us in comparing two or more groups or frequency distributions in teams of typical or characteristics performance.
1. Arithmetic Mean or Mean (M)
It is the simplest but most useful measure of central tendency.
It can be defined as the sum of all the values of the items in a series divided by the number of items.
It is denoted by symbol 'M'.
According to -
Ferguison - Mean is the sum of a set of measurement divided by the number of measurements in the set.
F.C. Mills - The arithmetic mean is the centre of gravity of a distribution.
A. Computation of mean in the case of ungrouped data -
M = ∑X / N
where, ∑X is the sum of values
and, N is the total number of items.
E.g. - If 3, 4, 5, 7, 11 are five numbers then the mean of these numbers will be -
Mean = M = (3 + 4 + 5 + 7 + 11) / 5
= 30/5
= 6
B. Computation of mean in the case of grouped data -
i) Direct method -
M = ∑fX / N
where, X is the mid-point of class
f is the respective frequency
N is the total of all frequencies
Let's understand this with an example -
M = ∑fX / N
= 2230/50
= 44.6
ii) Short Cut method -
M = A + (∑fd / N) × i
where, A is Assumed Mean
f is the respective frequency
N is the total of all frequencies
i is the class interval
and, d = (X - A) / i
X is the mid-point of the class
Let's understand this with an example -
Let -
Assumed mean (A) = 42
Class interval (i) = 5
Usually the assumed mean is the mid-point of class with the maximum frequency.
M = A + (∑fd / N) × i
= 42 + 26/50 × 5
= 42 + 2.6
= 44.6
2. Median (Md)
If the items of a series are arranged in ascending or descending order of magnitude, the measure or value of the central item in the series is termed the median.
According to -
Bloomers and Lindquist - The median of a distribution is the point on the score scales below which one half or 50 percent of the scores fall.
A. Computation of median in case of ungrouped data -
i) When the number of items (N) in the series are odd -
Mf = (N+1)/2 th term
E.g. - Median of 17, 47, 15, 35, 25, 29, 39
Numbers in ascending order = 15, 17, 25, 29, 35, 39, 47
Median = Md = (7 + 1) / 2
= 4th term
= 29
ii) When the number of items (N) in the series are even -
Md = [N/2 th term + (N/2 + 1) th term] / 2
E.g. - Median of 17, 47, 15, 35, 39, 50, 40, 44
Numbers in ascending order = 15, 17, 35, 39, 40, 44, 47, 50
Median = Md = [8/2 th term + (8/2 + 1) th term] / 2
= (4th term + 5th term)/2
= (39 + 40) / 2
= 39.5
B. Computation of median in case of grouped data -
Md = L + [(N/2 - cfb) / fm] × i
where, L is exact lower limit of median class
N is total of all frequencies
cfb is cumulative frequency below the median class
fm is frequency of median class
i is the class interval
Let's understand this with an example -
Here, Median (Md) = N/2 th term = 60/2 th term = 30 th term
and, median class = 50-59 (30th term falls in this class)
Median = Md = L + [(N/2 - cfb) / fm] × i
= 49.5 + [(30 - 16) / 20] × 10
= 49.5 + 7
= 56.5
There may arise a special situation where there are no cases within the interval containing the median. Let's understand this situation with an example.
This method is applicable only if the value of N/2 is equal to the value of cumulative frequency of the class with zero cases.
The class interval with zero frequency which is affecting calculations is adjusted towards the adjoining class interval on either sides and the size of modified class interval is used while applying the formula. The modified class intervals are mentioned in teams of exact limits.
Median = Md = L + [(N/2 - cfb) / fm] × i
= 10.5 + [(17 - 8) / 9] × 4.5
= 10.5 + 4.5
= 15
3. Mode (Mo)
It is the number which occurs most frequently in a series.
According to -
Crow and Crow - The score in a given set of data that appears most frequently is called mode.
Kenny - Mode is the most frequently occuring value in the series.
A. Computation of mode in case of ungrouped data -
Mode of 8, 9, 9, 13, 14, 13, 17, 16, 17, 16, 10, 20, 17
Here 17 is repeated a maximum of 3 times.
Mode = Mo = 17
B. Computation of mode in case of grouped data -
i) Indirect method -
Mode = 3 Median - 2 mean
ii) Direct method -
Mo = L + [(f1 - f0) / (2 f1 - f1 - f2)] × i
where, L is actual lower limit of modal class
f1 is frequency of modal class
f0 is frequency of class before modal class
f2 is frequency of class after modal class
i is the class interval
Let's understand this with an example -
Modal class is the class having highest frequency = 50-59
Mode = Mo = L + [(f1 - f0) / (2 f1 - f1 - f2)] × i
= 49.5 + [(10 - 6) / (20 - 6 - 7)] × 10
= 49.5 + 5.71
= 55.21
It may be possible to have more than one mode in a distribution, whereas, it is not possible in case of mean and median.
Mode is only a rough estimate, it can never be an accurate measure.