Introduction to Mean: Measure of Central Tendency
The Measure of Central Tendency, Mean (known as average or arithmetic mean) is used to describe the data set as a single number (value). The measure of central tendency represents the middle (center) of the data, that is the average measure (performance or behaviour, etc) of data. This measure of central tendency is also known as a measure of central location or measure of center.
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Mathematically, the mean can be defined as the sum of all values in a given dataset divided by the number of observations in that data. The mean is also called arithmetic mean or simply average.
Example of Mean
Example: Consider the following data set consisting of the marks of 15 students in a certain examination.
50, 55, 65, 43, 78, 20, 100, 5, 90, 23, 40, 56, 70, 88, 30
The mean of the above data values is computed by adding all these values (50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 = 813) and then dividing by the number of observations added (15) which equals 54.2 marks, that is
$\frac{50 + 55 + 65 + 43 + 78 + 20 + 100 + 5 + 90 + 23 + 40 + 56 + 70 + 88 + 30 }{15}=\frac{813}{15}=54.2 $
Statistical Representation of Population Mean
The above procedure of calculating the mean can be represented mathematically
$\mu= \frac{\sum_{i=1}^n X_i}{N} $
The Greek symbol $\mu$ (pronounced “mu”) represents the population mean in statistics and $N$ is the number of observations in the population data set.
The above formula is known as the population means as it is computed for the whole population.
Statistical Representation of Sample Mean
The sample mean can also be calculated in the same manner as the population mean is calculated. Only the difference is in the representation of the formula, that is,
$\overline{X}= \frac{\sum_{i=1}^n X_i}{n} $.
The $\overline{X} $ represents the sample mean and $n$ shows a number of observations in the sample.
Points to Remember When Calculating Mean
The mean is used for numeric data only. In Statistics, calculating the mean, the data type should be Quantitative (measured on either a ratio or interval scale). Therefore, the numbers in the data set can be continuous and/ or discrete.
Note that mean should not be computed for alphabetic or categorical data (should not belong to nominal or ordinal scale). The mean is influenced by extreme values (very very large or small value) in data that change the mean drastically.
For other measures of central tendencies visit: Measures of Central Tendencies
