Objective

This topic gives an overview of;

• Representative Values
• Arithmetic Mean
• Range
• Mode
• Median

Representative Values

Average is a number that represents or shows the central tendency of a group of observations or data. Since average lies between the highest and the lowest value of the given data so, we say average is a measure of the central tendency of the group of data.

Let us take an example; the average temperature at this time of the year is about 40 degree Celsius. Here the average temperature of 40 degree Celsius means that, very often, the temperature at this time of the year is around 40 degree Celsius. Sometimes, it may be less than 40 degree Celsius and at other times, it may be more than 40 degree Celsius.

Different forms of data need different forms of representative or central value to describe it. One of these representative values is the “Arithmetic Mean”.

Arithmetic Mean

The most common representative value of a group of data is the arithmetic mean or the mean. The arithmetic mean of a set of data is found by taking the sum of the data, and then dividing the sum by the total number of values in the set. A mean is commonly referred to as an average.

The average or Arithmetic mean (A.M.) or simply mean is defined as follows:

Let us consider an example Two vessels contain 20 litres and 60 litres of milk respectively. The question is to find the amount that each vessel would have, if both share the milk equally.

Thus the average or the arithmetic mean would be,

Thus, each vessel would have 40 litres of milk.

Range

The difference between the highest and the lowest observation gives the Range of the observation. Let us consider an example. In {4, 6, 9, 3, 7} the lowest value is 3, and the highest is 9, so the range is 9-3 =6.

Mode

Mode is the most frequently occurring value in a frequency distribution. For different requirements from a data other measures like Mode of Central tendencies are used. The mode of a set of observations is the observation that occurs most often.

The following are the steps to calculate mode:

•  Arrange the data in ascending order.

•  Tabulate the data in a frequency distribution table.

•  The most frequently occurring observation will be the mode.

Let us consider an example. In the given set {1, 1, 2, 4, 3, 2, 1, 2, 2, 4}, by arranging them in ascending order we get 1, 1, 1, 2, 2, 2, 2, 3, 4, 4 .Thus the mode of this data is 2 because it occurs more frequently than other observations.

For large data putting the same observations together and counting them is not easy if the number of observations is large. In such cases we tabulate the data. Tabulation can begin by putting tally marks and finding the value which has occurred highest number of times.

Median

Median refers to the value which lies in the middle of the data (when arranged in an increasing or decreasing order) with half of the observations above it and the other half below it. We consider only those cases where number of observations is odd. Thus, in a given data, arranged in ascending or descending order, the median gives us the middle observation.

The following are the steps to calculate Median:

• Arrange the data in ascending order.

•  The value that lies in the middle such that half of the observations lie above it and the other half below it will be the median.

Let us consider an example. In the given set {24, 36, 46, 17, 18, 25, 35}, by arranging them in ascending order we get 17, 18, 24, 25, 35, 36, 46. Median is the middle observation. Therefore 25 is the median.

Thus we realise that mean, mode and median are the numbers that are the representative values of a group of observations or data. They lie between the minimum and maximum values of the data. They are also called the Measures of the Central Tendency.

Summary

• Average is a number that represents or shows the central tendency of a group of observations or data.

• Arithmetic mean is one of the representative values of data.

• The difference between the lowest and highest values is called Range of values.

• Mode is another form of central tendency or representative value. The mode of a set of observations is the observation that occurs most often.

• Median is also a form of representative value. It refers to the value which lies in the middle of the data with half of the observations above it and the other half below it.