CAT / FIA FMA Syllabus B. Data Analysis And Statistical Techniques - Measures of average and dispersion - Notes ^{5 / 7}

Variables can be:

• Discrete variables which can only consist of certain values (ungrouped data), for example a company recording the number of complaints received per week over the last year.

• Continuous variables where we group the variables, for example a company recording the total amount paid to employees each week over the last year.

Measures of average for ungrouped data

There are 3 measures of average that we need to be aware of for ungrouped data:

1. Arithmetic mean

   This is calculated by adding up all of the observations and dividing by the number of observations

2. Median

   This is the centrally occurring observation when all of the observations are arranged in order of magnitude

3. Mode

   This is the most frequently occurring observation

Illustration 1

A company has recorded the number of complaints received per week over the past thirteen weeks, and has produced the following table:

Calculate: 

1) The arithmetic mean
2) The median
3) The mode

Solution

The mean is: 20/13 = 1.54 (table below)

The median is 1, as this the most centrally occurring observation when all of the observations were arranged in order of magnitude (table below)

The mode is 1, as this is the most frequently occurring observation.

Measures of average for grouped data

We need to be aware of how to calculate the arithmetic mean for grouped data, to do this we must:

1) Find the mid point of our observations
2) Total mid points (mid point x number of times it was observed)
3) Divide by the number of observations.

Illustration 2

A company has recorded the total amount paid to employees each week over the last year in the following table:

Calculate the arithmetic mean

Solution

The arithmetic mean = 94,500 / 52 = $1,817 (table below)

Measures of dispersion

Dispersion is looking at the spread of the observations. 

We need to know the following measures of dispersion:

• Range

  This is the difference between the highest and lowest of the observations

• Variance

  The variance is calculated as follows:

  1) Find the difference between the observation and the arithmetic mean (For ungrouped data we will use the mid point observation)
  2) Square the difference
  3) Multiply the squared difference with the number of times that it occurred and find the total
  4) Take an average of this total

• Standard Deviation

  This is the square root of the variance

• Coefficient of variation

  This is the standard deviation divided by the arithmetic mean

Illustration 3

Using Illustration 1 find the:

1) Range
2) Variance
3) Standard deviation
4) Coefficient of variation

• Solution

  Range 3 - 0 = 3 (Highest value - lowest value)

• The variance (W1) = 9.21/13 = 0.71

• Standard deviation √ 0.71 = 0.84

• Coefficient of variation = 0.84/1.54 = 0.55

  W1: