Statistics

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Chapter 4

Numerical descriptive measures

Numerical Descriptive Techniques…

• Measures of central location

Mean, median, mode

• Measures of variability

Range, standard deviation, variance, coefficient of variation

• Measures of relative standing

Percentiles, quartiles

• Measures of linear relationship

Covariance, correlation, determination, least squares line

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4.1 Measures of central location

Three main types of measures of central location are: • Arithmetic mean (or average) • Median • Mode

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Arithmetic Mean (or Average)

This is the most popular and useful measure of central location. Mean =

Sample mean

Sum of measurements Number of measurements

Population mean

Sample size

Population size

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Example 4.1

The mean of the sample of six measurements 1, 3, 5, 2, 4, 3 is given by 1 3 5 2 4 3 3.0

Example

Suppose the telephone bills of Example 2.5 represent a population of measurements. The population mean is 42.19 38.45 53.21 43.59

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Example

When many of the measurements have the same value, the measurement can be summarised in a frequency table. Suppose the numbers of children in a sample of 20 families were recorded as follows: NUMBER OF CHILDREN 0 1 2 3 4 NUMBER OF FAMILIES 3 4 7 2 4

20 families

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The Arithmetic Mean…

• …is appropriate for describing measurement

data, e.g. heights of people, marks of student papers, etc.

• …is seriously affected by extreme values called ‘outliers’. E.g. as soon as a billionaire moves into a neighborhood, the average household income increases beyond what it was previously!

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Median

• Another most commonly used measure of central location is the median. • The median of a set of measurements is the value that falls in the middle when the measurements are arranged in order of magnitude.

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Median

Example 4.2

Seven employee salaries were recorded (in 1 000s) : 42, 45, 40, 46, 44, 40, 43....