Standard Deviation

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Standard Deviation: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Standard deviation is a number describing how far the values in a data set typically sit from their mean, so a small one means tightly clustered data.

Algebra 1StatisticsS-ID.A.2

What is Standard Deviation?

Two classes can both average 80 on a quiz, yet in one every score hugs 80 while the other runs from 50 to 100, and the standard deviation captures that difference with one number, a typical distance from the mean.

To compute it for a full set of \(n\) values: find the mean, subtract it from each value, square those deviations, average the squares to get the variance, then take the square root. A set whose values are all identical has a spread of zero, and a larger result means a wider scatter. For a sample, statisticians divide by \(n – 1\) instead of \(n\).

How to spot it

  • A question compares how consistent two data sets are, even though their averages match.
  • The wording mentions spread, variability, typical distance from the mean or consistency.
  • A calculator or spreadsheet output includes a symbol like \(\sigma\) or s.
  • You are told to describe a data set with both a center and a measure of spread.

The method

  1. Find the mean of the data.
  2. Subtract the mean from every value, then square each difference.
  3. Add the squares and divide by n (the count) to get the variance.
  4. Take the square root of the variance to land back in the original units.
3456789101112mean = 7deviations: −3, −1, −1, 1, 4squares: 9 + 1 + 1 + 1 + 16 = 2828 ÷ 5 = 5.6, so σ = √5.6 ≈ 2.37
Data 4, 6, 6, 8, 11 have mean 7 and standard deviation about 2.37.

Standard Deviation: solved example

Problem

Weekly walking times to class, in minutes, were 12, 15, 15, 18 and 20. Find the standard deviation, treating these as the whole set.

Solution

The mean is \(\frac{12 + 15 + 15 + 18 + 20}{5} = 16\). The deviations are \(-4, -1, -1, 2, 4\), and their squares are \(16, 1, 1, 4, 16\).

The squares total 38, so the variance is \(\frac{38}{5} = 7.6\), and \(\sqrt{7.6} \approx 2.76\). The typical distance from the mean is about 2.76 minutes.

Try it

Question 1

What is the standard deviation of 10, 10, 10, 10?

Show the answer

Zero: every value equals the mean, so there are no deviations.

Question 2

Which set is more spread out, \(\{5, 5, 6, 6\}\) or \(\{1, 3, 9, 11\}\)?

Show the answer

The second; its values lie far from the mean of 6, so its deviation is larger.

Standard Deviation vs. Mean Absolute Deviation

Mean absolute deviation averages the plain distances from the mean. This measure squares the distances first and takes a root at the end, so a few far-off values count for more. Both describe spread, but they usually give different numbers for the same data.

Mean Absolute Deviation (MAD)

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