What is the value of the middle observation when the observation are arranged in the order of their magnitude?

A population is the whole set of items from which a data sample can be drawn; so a sample is only a portion of the population data.  Features of a sample are described by statistics.

Statistical methods enable us to arrange, analyse and interpret the sample data obtained from a population.  We gather sample data when it is impractical to analyse the population data as a smaller sample often allows us to gain a better understanding of the population without doing too much work or wasting precious time.

Measures of Central Tendency

We make inferences about a population from a sample set of observed values by finding the mean, median and mode.  The mean, median and mode are collectively known as measures of central tendency.

Mean

The mean (or average) of a set of values is defined as the sum of all the values divided by the number of values.  That is:

Example 1

The marks of five candidates in a mathematics test with a maximum possible mark of 20 are given below:

14     12     18     17     13

Find the mean value.

Solution:

So, the mean mark is 14.8

Median

E.g. the median of 3, 5 and 9 is 5 as it is the middle value.

Example 2

The marks of five candidates in a geography test for which the maximum possible mark was 20 are given below:

18     17     15     14     19

Find the median mark.

Solution:

Arrange the marks in ascending order of magnitude:

14     15     17     18     19

The third value, 17, is the middle one in this arrangement.

If the number of values in the data set is even, then the median is the average of the two middle values.

Example 3

Find the median of the following scores:

10     16     14     19     8     11

Solution:

Arrange the values in ascending order of magnitude:

8     10     11     14     16     19

There are 6 values in the data set.

The third and fourth values, 11 and 14, are in the middle.  That is, there is no one middle value.

Half of the values in the data set lie below the median and half lie above the median.

Mode

E.g. the mode of the data set {3, 5, 6, 7, 7, 7, 8, 8, 9} is 7 as it occurs most often.

Example 4

The marks awarded to seven pupils for an assignment were as follows:

18     14     18     15     12     19     18

a.  Find the median mark.
b.  State the mode.

Solution:

a.  Arrange the marks in ascending order of magnitude:

12     14     15     18     18     18     19

The fourth mark, 18, is the middle data value in this arrangement.

b.  18 is the mark that occurs most often.

• The mode has applications in manufacturing.  For example, it is important to manufacture more of the most popular cars, because manufacturing different cars in equal numbers would cause a shortage of some cars and an oversupply of others.
• The mean (or average) of a set of data values is used to predict future results.
• The median is easy to comprehend.  Extreme values in the data set do not affect the median.

statistics, population, sample, measures of central tendency, mean, average, median, mode

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