What is the Real Meaning of Average in Statistics?
Average is a single value which is calculated to represent the whole data. It may be calculated for Sample or Population data.
It is a single value that represents whole data. It is a value somewhere in the center, where most of the items of the series cluster. Such values are called Measures of central tendency. A choice of the proper average is the job of an expert who is calculating the average.
Types of Average in Statistics
The following averages are usually used:
- Arithmetic mean
- Geometric mean
- Harmonic mean
- Median
- Mode
What is Arithmetic Mean in Statistics?
Arithmetic mean is the sum of values, divided by the total number of values. it is also called X bar. It is obtained by adding up all observation and dividing by total number of observations.
Arithmetic Mean Formula for Grouped and Ungrouped Data
Arithmetic Mean formula for Ungrouped data
\frac{Ex}n
Arithmetic Mean formula for Grouped data
\frac{Efx}{Ef}
Properties of Arithmetic Mean as a Measure of Central Tendency
It is most widely used measure of location. Its properties are:
- A set of numerical data has one and only one mean, so it is unique.
- All the values are included in computing arithmetic mean
- The sum of squared deviation from mean is minimum.
Demerits of Arithmetic mean
- It cannot average ratios and percentage properly.
- it can not be computed if any item is missing.
- it is highly affected by extremely large values.
Geometric Mean Examples with Solutions
The Geometric mean of a set of data of n positive number is the nth root of their product. It is obtained by multiplying all the values and then extracting the relavent root of the product.
Properties of Geometric Mean
- The Geomatric mean in the terms of A.M and H.M is:
- G.M = \sqrt{A.M\;\times H.M}
- The geometric mean is always less then arithmetic mean.
- It is considered best tool for constructing index number.
- Geometric mean is used for calculation of average percentage increase or decrease.
Demerits of Geometric Mean
- its calculation is rather difficult.
- it is not easy to understand.
- it cannot be calculated if any item is zero or negative.
What is Harmonic mean?
Harmonic mean(H) is defined as the number of values, divided by sum of reciprocal of each value. Or Harmonic mean of series is the reciprocal of the arithmetic mean of the reciprocal of the values.
Harmonic mean Formula
H.M\;=\;\frac n{{\displaystyle\frac1{x_1}}+{\displaystyle\frac1{x_2}}+\;{\displaystyle\frac1{x_3}}+…{\displaystyle\frac1{x_n}}\;}
Properties of Harmonic mean
- It is used to measure average speed, average price , average of profit and loss.
- Sometimes Harmonic mean is used as an alternative to weighted arithmetic mean.
Demerits of Harmonic mean
- its calculation is rather difficult.
- it gives high weightage to small values.
- it is usually a value that does not exist in given data.
Define Median
The median is the middle value of set of Sara that are arranged in ascending or descending order of magnitude. If sample size n is an odd number then median is middle value. If sample size n is an even number, the median is average of two middle values. Median divides the data into two equal parts. It can also be called X_tilde.
Properties of Median
- Median is unique, there is only one median in a set of data.
- It is not affected by extremely large values or small values.
- The median is used for an Open_ended distribution.
Demerits of Median
- it is not capable for further mathematical treatments.
- it cannot give correct value when multiplied by number if items.
Define Mode
The mode is the value in a set of Sara that appears maximum number of times. In the other words mode is the value that appears with the highest frequency in a set of data.
Properties of Mode
- The mode is very easy to find and thus is used to guide to the typical values in the sample.
- It can be found for quantitative data as well as qualitative data.
- The mode is not affected by occurrence of any extreme value.
Demerits of Mode
- it has no significance when number of items is not large.
- it is not based on all observations.
- it is not capable for further mathematical treatments.
EMPIRICAL RELATION BETWEEN MEAN ,MEDIAN , MODE
- Mean – median = \frac12 (median – mode)
- Mean – mode= 3 (mean – median)
- Median – mode= 2(Mean- Median)
OR
- Mean =\frac12 (3 median – mode)
- Median = \frac13 (2 mean + mode)
- Mode = 3 median – 2 Mean
Relationship between A.M ,G.M ,H.M:
if all observations are same then:
A.M=G.M=H.M