Measures of Central Tendency
A measure of central Tendency is a value that gives the central position of given data. Measures of central tendency condensing the mass of data in one single value. Central tendency enable us to get an idea of the entire data.
Properties of a Good Measure of Central Tendency:
• It should be easy to understand and calculate.
• It should be rigidly defined.
• It should be based on all observations.
• It should be least affected by sampling fluctuation.
• It should be capable of further algebraic treatment.
• It should be least affected by extreme values.
Following are three important measures of central tendency which are commonly used
I. Arithmetic Mean
II. Median
III. Mode
| Central Tendency | Definition | Formula for Ungrouped Data | Formula for Discrete Data | Formula for Continuous Data |
|---|---|---|---|---|
| Arithmetic Mean or Mean (x̄) | The arithmetic mean is defined as sum of the numerical values of each and every observation divided by the total number of observations |
x̄ = (1/n) Σi=1n Xi n = Number of observations |
X̄ = ΣXifi/N N = Total frequency |
X̄ =Σ Xifi /N Xi = middle value of class interval N = Total frequency |
| Median (M) | Middle most or central value of observation is called as median. Median devides the given data into two equal parts. |
I] When n is odd number: M{(n+1)/2}th term II] When n is even number: M = [ (n/2) + (n+1)/2 ] / 2 |
M = (N/2)th term N = Total Frequency |
M = l + [(N/2 – C) / f] × h Where: l = Lower limit of class interval of median f = Frequency of class where median lies C = Cumulative frequency before median class N = Total frequency h = Width of class interval |
| Median (M) | Middle most or central value of observation is called as median. Median devides the given data into two equal parts. |
I] When n is odd number: M={(n+1)/2}th term II] When n is even number: M = [ (n/2) + (n+1)/2 ] / 2 |
M =(N+1)/2th term N = Total Frequency |
M = l + [(N/2 – C) / f] × h Where: l = Lower limit of median class CI f = Frequency of median class CI C = Preceding cumulative frequency of median class CI N = Total Frequency h = Width of CI |
| Mode (Mo) | Most occurring value or most repeating value of observation is called as mode. | Most repeating value of observation | The observation which corresponds to maximum frequency. |
Mo = l + [(fm – f1) / (2fm – f1 – f2)] × h Where: l = Lower limit of mode class CI fm = Frequency of mode class CI f1 = Preceding Frequency of mode class CI f2 = Succeeding Frequency of mode class CI h = Width of mode class CI |
