Math's: Measures of Central Tendency: Mean and Calculation of Mean
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Measures of Central Tendency- Meaning
A representative central value for a large data is known as a measure of central tendency.
Mean, Median and Mode are the three most common measures of central tendency.
Arithmetic Average or Mean
The mean of a data is the central value based on all the observations in a data series. Mean is denoted by .
Calculation of Mean
Individual Series
Individual series is a data series in which all the observations occur once i.e. all observations have frequency 1.
Mean in individual series is found out by adding all the observations in a data series and dividing the sum by the numbers of observations in the series. Thus, the mean of observations is .
Example – The weight of four bags of rice are 120 kg, 100 kg, 80 kg and 110 kg. Find the mean weight.
Sol- Mean Weight
Ungrouped Frequency Series or Discrete Frequency Series
Ungrouped Frequency Series is the series in which the observations are not grouped into classes and some or more observations occur more than once.
In ungrouped frequency series, we first find by multiplying each observation with its respective frequency and then divide the summation of or by the total of frequencies .
Thus,
Short Cut Method
Here, we take a constant as an assumed mean and then find the deviations of each observation from it . The deviations are multiplied with respective frequencies to get . Then, the mean is obtained using the following formula-
Example- The marks obtained out of 10 by twenty students in a class test are as follows-
Marks Obtained (X) | No. of Students (f) |
10 | 4 |
8 | 12 |
6 | 4 |
Find the mean marks.
Solution-
Marks Obtained | No. of Students (f) | |
10 | 4 | 40 |
8 | 12 | 96 |
6 | 4 | 24 |
Grouped Frequency Series or Interval Series
In Interval Series, we have various class intervals with a range of values for data intervals with their respective class frequencies.
For calculating mean of an interval series, we go through the following steps-
Calculate the mid values of the class intervals by working out the average of upper limit and lower limit of the class.
Multiply the mid values with the respective frequencies of class to get
Compute and divide it with the sum of frequencies to get the mean value.
Short Cut Method
This is similar to that in discrete series. Here, we take the deviations from the mid values and apply the following formula-
Example- Find the mean of the following distribution
Class | Frequency |
10-12 | 2 |
12-14 | 5 |
14-16 | 3 |
Solution
Class | Frequency (f) | Mid –Values (m) | |
10-12 | 2 | 11 | 22 |
12-14 | 5 | 13 | 65 |
14-16 | 3 | 15 | 45 |