UNIT 14 : Measures of dispersion
Key unit competence
Extend understanding, analysis and interpretation of data arising from problems and questions in daily life to include the standard deviation.
Learning objectives
14.1 Introduction
In Junior Secondary, you were introduced to statistics. You learnt about measures of central tendencies. In this unit, we shall learn about measures of dispersion.
Activity 14.1
Discuss in groups the meaning of measures of central tendencies. What are they? Where can we apply them?
Definitions
A measure of central tendency; also called average, is values about which the distribution of data is approximately balanced. There are three types of measure of central tendency namely the mean, the median and the mode.
Mean: is the sum of data values divided by the number of values in the data.
Mode: is the value that occurs most often in the data.
Median: is the middle value when the data is arranged in order of magnitude.
The mean
The median
The median of data is the middle value when all values are arranged in order of the size.
When the number of the items is odd then the median is the item in the middle. If and when the number of items is even, the median is the mean of the two numbers in the middle.
The mode
The mode of a set of data is the value of the higher frequency in the distribution of marks
In Example 14.3 the mode is 45 because it has the highest frequency, 12.
Grouped data
Grouped data is commonly used in continuous distribution data that takes any value in a given range is called continuous data.
Such data has values which are only approximations, such as height, weight, mass, time, age and temperature.
The mean of grouped data
In order to find the mean of the grouped data:
The mean from an assumed mean
When data is grouped in classes of equal width, we use the assumed mean in order to reduce the numerical size of the value of n.
Determine the mid-point of each class interval and the classes of the central value of x which is usually the modal value. This value is referred to as an assumed value or working mean.
14.2 Measures of dispersion
Activity 14.2
In groups of three, research on the meaning and types of measures of dispersion. Why are they useful? Discuss your findings with the rest of the class.
A measure of dispersion is the degrees of spread of observation in data. The common measure of dispersion are range inter-quartiles, range and the standard deviation (the square root of the variance)
Range
Inter-quartile range
The inter-quartile range is the central 50% of a distribution when it is arranged in order of size. It is given by the formula Q3 – Q1 where Q1 is the lower quartile and Q3 the upper quartile.
Semi inter-quartile range
The semi-interquartile range (or SIR) is defined as the difference of the first and third quartiles divided by two
The standard deviation and the variance
Alternative form of the formula for standard deviation
14.3 Coefficient of variation
14.4 Application
Activity 14.3
Carry out research to find out real life applications on measures of dispersion. Discuss your findings with the rest of the class.
