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### General

### 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.

**Supplementary, interactive questions served by Siyavula Education.**Note: Questions will open in a new window or tab.

**Introduction**- The mode toad
- The mode
- The median
- What does mode mean?
- The mean
- Truths about mean, median & mode
- Measures of central tendency
- What is it: mean, median, or mode?
- Working with the mean
- Definition: modal interval
- Mean calculations for grouped data
- Finding the modal interval for grouped data
- The estimated mean compared to the true mean
- Where is the median?
- Estimating the mean for grouped data

**Supplementary, interactive questions served by Siyavula Education.**Note: Questions will open in a new window or tab.

**Measures of dispersion**- The range
- Which is which: mean, median, mode & range
- Dispersion & central tendency
- Calculating the range of a data set
- Percentile and quartiles
- Working with the range of a data set
- Percentile and rank in a data set
- Finding the range by solving an equation
- Applying percentile to a data set
- Comparing how values are spread out
- Standard deviation values
- Mode, range & standard deviation
- Roll the dice
- Calculating variance
- What does sigma = 0 mean?