Some aspects of the data can be described quantitatively to represent certain features of the data. An average is one of such representative measures.
As average is a number of indicating the representative or central value of the data, it lies somewhere in between the two extremes. For this reason, average is called a measure of central tendency.
Central Tendency: A single quantity which enables us to know the average characteristics of the data under consideration. Use of central tendency is a technique to analyse the data.
Various Measures of Central Tendency
- Arithmetic mean or mean or average
- Median
- Mode
Mean
It is the ratio of the sum of all values of the variable and the number of observations.
Mean of Raw Data
To calculate the mean of raw data, all the observations of the data are added and their sum is divided by the number of observations.
$$ \overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} $$
$$ \overline{x} = \frac{1}{n} \sum_{i=1}^{n} x_{i} $$
Example 1: The weight of four bags of wheat (in kg) are 103, 105, 102, 104. Find the mean weight.
Example 2: The enrolment in a school in last five years was 605, 710, 745, 835 and 910. What was the average enrolment per year?
Example 3: The mean of marks obtained by 30 students of Section A of Class X is 48, that of 35 students of Section B is 50. Find the mean marks obtained by 65 students in Class X.
Mean marks of 30 students of Section A = 48
So, total marks obtained by 30 students of Section A = 30 × 48 = 1440
Similarly, total marks obtained by 35 students of Section B = 35 × 50 = 1750
Total marks obtained by both sections = 1440 + 1750 = 3190
Mean of marks obtained by 65 students = 3190/65 = 49.1 approx.
Example 4: The mean of 6 observations was found to be 40. Later on, it was detected that one observation 82 was misread as 28. Find the correct mean.
Mean of 6 observations = 40
So, the sum of all the observations = 6 × 40 = 240
Since one observation 82 was misread as 28, therefore, correct sum of all the observations = 240 - 28 + 82 = 294
Hence, correct mean = 294/6 = 49
Mean of Ungrouped Data
$$ \overline{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}} $$
To find mean, first find fixi, by multiplying each xi with its corresponding frequency fi. Append a column of fixi in the frequency table.
Assumed Mean Method
Sometimes when the numerical values of xi and fi are large, finding the product fi and xi becomes time consuming. We choose an arbitrary constant a, also called the assumed mean and subtract it from each of the values xi. The reduced value, di = xi - a is called the deviation of xi from a.
This method of calculation of mean is known as Assumed Mean Method.
Mean of Grouped Data
Frequency in any class is centred at its class mark or mid point.
Median
There is a weakness of the mean. It is affected by the extreme values of the observations in the data. This weakness of mean drives us to look for another average which is unaffected by a few extreme values. Median is one such a measure of central tendency.
Median is a measure of central tendency which gives the value of the middle-most observation in the data when the data is arranged in ascending (or descending) order.
Median of Raw Data
When the number of observations (n) is odd, the median is the value of (n+1)/2 observation.
When the number of observations (n) is even, the median is the mean of the (n/2) and (n/2 + 1) observations.
Mode
The observation that occurs most frequently in the data is called mode of the data. It is an observation with the maximum frequency in the given data.