Standard Deviation

The Variance and the Standard Deviation are measures of spread of the data indicating how far the data is away from the normal (mean).

When data such as people's heights, weights, intelligence, school test marks and so forth are graphed, they tend to create a "bell-shaped curve" or normal distribution curve.

Mathematical Symbols:

Normal Distribution Graph:

Standard Deviation diagram

A Normal Distribution curve with sufficient data has the following features:


Finding the Standard Deviation of a Population


Example One - Intelligence Quotient (IQ)

Einstein and Hawking Portraits


IQ Bell Curve

In most IQ tests, the "average" or Mean IQ is about 100 and the Standard Deviation is about 15. The vast majority of people have IQs between 70 and 130. However, famous physicists Albert Einstein and Stephen Hawking (in the photographs above) are thought to have IQs of about 145 (which is equal to the Mean plus 3 Standard Deviations). Refer to the instructions above to calculate the mean, the variance and the standard deviation of the IQ data of the following fictitious people listed in this table.

PERSONIQ
Al76
Bert80
Cuc89
David96
Eng100
Fran102
Gianni106
Helena108
Ines122
Jack130

Answer:

PERSONIQIQ DIFFERENCE FROM THE MEAN SQUARE OF THE IQ DIFFERENCE FROM THE MEAN
Al76– 24.9620.01
Bert80– 20.9436.81
Cuc89– 11.9141.61
David96– 4.924.01
Eng100– 0.90.81
Fran102+ 1.11.21
Gianni106+ 5.126.01
Helena108+ 7.150.41
Ines122+ 21.1445.21
Jack130+ 29.1846.81
MEAN IQ = 100.9MEAN OF SQUARED DIFFERENCES = 259.29

Mean = 100.9
Variance = 259.29
Standard Deviation = square root of variance = √ 259.29 = 16.1


Example Two - World's Tallest Structures

STRUCTUREHEIGHT IN METRES
Great Pyramid of Giza139
Eiffel Tower300
Empire State381
Petronas Towers452
Taipei 101509
World Trade Centre526
CN Tower553
KVLY-TV Mast629
Warsaw Radio Mast646
Burj Khalifa830

Tallest Buildings diagram Tallest Buildings diagram

The graph and the table show the heights of ten of the world's tallest structures. Draw a table and then calculate the:
(a) mean
(b) variance
(c) standard deviation.


Answer:

STRUCTUREHEIGHT IN METRESHEIGHT DIFFERENCE FROM THE MEAN SQUARE OF THE HEIGHT DIFFERENCE FROM THE MEAN
Great Pyramid of Giza139– 357.5127 806.25
Eiffel Tower300– 196.538 612.25
Empire State381– 115.513 340.25
Petronas Towers452– 44.51 980.25
Taipei 101509+ 12.5156.25
World Trade Centre526+ 29.5870.25
CN Tower553+ 56.53 192.25
KVLY-TV Mast629+ 132.517 556.25
Warsaw Radio Mast646+ 149.522 350.25
Burj Khalifa830+ 333.5111 222.25
MEAN HEIGHT = 496.5 mMEAN OF SQUARED DIFFERENCES = 22 597.525

Mean = 496.5 m
Variance = 22 597.525
Standard Deviation = square root of variance = √ 22 597.525 = 150.32 m

Questions to Consider

Which building is closest to the "average" height of these ten structures?
Why are some of the differences from the mean positive and others negative?
Why are these differences squared?

Maths Fun

Find out how to get the Mean and the Standard Deviation on your calculator without using a table. The symbols are: