MTH 110 – INTRODUCTION TO

STATISTICS

Session 4 – Measures of Dispersion, Variance and Standard Deviation Instructor: Manos Takas Email: m.takas@cityu.gr

Range Look at these two sets of data: 2, 3, 4, 5, 6 3,2,3,5,13 -3,2,3,5,13 They both have a mean of 4. However, you can see that the first data set (21 is more spread out than the second data set. The mean doesn't tell you this. _ To represent the data more accurately, you need the mean plus a measure of the spread or dispersion of the data. One simple measure of dispersion is the range. The range of a set of data is the highest value minus the lowest value

In this case the range of set [1] is 6 - 2 = 4. The range of set [2] is 13 (-3) = 16. The range is easy to calculate, but there are other measures of spread that are more useful.

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Variance and Standard Deviation The mean of the deviations from the mean squared is called the variance, usually written as σ 2

The variance is calculated as:

σ2

=

1 ∑ ( x − x )2 n

2

Sometimes it is easier to calculate a variance using the alternative formula:

σ

⎛∑x⎞ 1 = ∑ x2 − ⎜ ⎜ n ⎟ ⎟ n ⎝ ⎠

2

Standard Deviation The standard deviation σ is defined as the square root of the variance

⎛ ∑ x2 ⎞ 1 1 2 σ= ∑ x − ⎜ n ⎟ = n ∑ ( x − x )2 ⎜ ⎟ n ⎝ ⎠

2

Standard Deviation for Frequency Distributions

σ2 =

1 ∑ n

⎛ ∑ fx ⎞ fx 2 − ⎜ ⎜ n ⎟ ⎟ ⎝ ⎠

2

Where f represents the frequency of the x observation and n=Σf

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