Standard Deviation & Variance

The variance is another measure for the spread of the data, it measures the variability from the mean of the data.

What is the variance and the standard deviation?

The variance is a statistic that tells us how varied a set of data is
- Data that is more spread out will have a greater variance
- Data that is consistent and close together will have a smaller variance
The standard deviation is the square root of the variance
The symbol for the population standard deviation is the lowercase Greek letter sigma, σ and for variance is sigma squared, σ²
Standard deviation and variance are used interchangeably within this course so make sure you look out for which one a question shows or asks for

How are the variance and standard deviation calculated?

There is more than one formula that can be used for calculating the variance, and you should choose the most useful one
For a set of $n$ values $x_{1}, x_{2}, . . ., x_{i}, . . ., x_{n}$ the variance is the sum of the squares of the deviations from the mean, divided by the frequency
- Variance = $\frac{Σ {(x - \bar{x})}^{2}}{n}$
  - This formula can be time consuming and therefore is rarely used in this statistics course
A second, easier to use version of the variance is:
- Variance = $\frac{Σ x^{2}}{n} - {\bar{x}}^{2}$
- This version is easier to work with and should be used in most instances
- Variance can also be written in other ways

Variance = $σ^{2} = \frac{Σ {(x - \bar{x})}^{2}}{n} = \frac{1}{n} (Σ x^{2} - \frac{{(Σ x)}^{2}}{n}) = \frac{Σ x^{2}}{n} - {(\frac{Σ x}{n})}^{2} = \frac{Σ x^{2}}{n} - {\bar{x}}^{2}$

- - An easy way to remember this is to think of it as ‘the sum of x squared over n minus the sum of the mean squared’
  - Most calculators can be used to find summary statistic such as the standard deviation and variance fairly quickly, practice finding it on yours

The standard deviation is the square root of the variance
- Standard deviation = $σ = \sqrt{\frac{Σ {(x - \bar{x})}^{2}}{n}} = \sqrt{\frac{Σ x^{2}}{n} - {\bar{x}}^{2}}$
  - Makes sure you know how to find these formulae in the formula booklet and are familiar with the version given
The units for standard deviation are the same as the units for the data and the units for variance are the same as the units for the data but squared

How are the variance and standard deviation calculated from a frequency table?

The method for finding the variance from a frequency table is similar to that of the mean
- If calculating from a grouped frequency table, find the midpoints, $x$ , first
- Multiply each $x$ value by its corresponding frequency and use these values within the formulae
- The formulae will become
  - Variance = $σ^{2} = \frac{Σ {(x - \bar{x})}^{2} f}{Σ f} = \frac{Σ x^{2} f}{Σ f} - {(\frac{Σ x f}{Σ f})}^{2}$
  - Standard deviation = $σ = \sqrt{\frac{Σ {(x - \bar{x})}^{2} f}{Σ f}} = {\sqrt{\frac{Σ x^{2} f}{Σ f} - (\frac{Σ x f}{Σ f})}}^{2}$

Worked example

Phoom recorded the length of time, $t$ , it took him, in minutes, to answer a selection of further calculus exam questions. The data is summarised in the table below.

Time (minutes)	Frequency
$2 \leq t < 4$	1
$4 \leq t < 6$	4
$6 \leq t < 8$	7
$8 \leq t < 10$	5
$10 \leq t < 12$	2
$12 \leq t < 20$	2

(i)

Calculate an estimate of the variance of the time taken to complete a question, include units with your answer.

(ii)

Write down an estimate of the standard deviation of the time taken to complete a question, include units with your answer.

2-1-3-sd-and-variance-we-solution-part-1

2-1-3-sd-and-variance-we-solution-part-2

Examiner Tip

Look out for whether a question gives or asks for the standard deviation or variance, especially if the question is using sigma notation.
Choose which formula to use wisely, most of the time the summary statistics will be given so only one of the formulae will be possible. On the rare occasion that you are asked to calculate directly from a table think carefully about which version of the formula is quickest and easiest to use. It will almost always be the second version given in this revision note.

Standard Deviation & Variance (CIE AS Maths: Probability & Statistics 1)

Revision Note

What is the variance and the standard deviation?

How are the variance and standard deviation calculated?

How are the variance and standard deviation calculated from a frequency table?

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1. Data Presentation & Interpretation

1.1 Statistical Measures

1.1.1 Basic Statistical Measures

1.1.2 Frequency Tables

1.1.3 Standard Deviation & Variance

1.1.4 Coding

1.2 Representation of Data

1.2.1 Data Presentation

1.2.2 Stem and Leaf Diagrams

1.2.3 Box Plots & Cumulative Frequency

1.2.4 Histograms

1.3 Working with Data

1.3.1 Interpreting Data

1.3.2 Skewness

2. Probability

2.1 Basic Probability

2.1.1 Calculating Probabilities & Events

2.1.2 Venn Diagrams

2.1.3 Tree Diagrams

2.2 Permutations & Combinations

2.2.1 Arrangements & Factorials

2.2.2 Permutations

2.2.3 Combinations

2.3 Further Probability

2.3.1 Set Notation & Conditional Probability

2.3.2 Further Tree Diagrams

2.3.3 Further Venn Diagrams

2.3.4 Probability Formulae

3. Statistical Distributions

3.1 Probability Distributions

3.1.1 Discrete Probability Distributions

3.1.2 E(X) & Var(X) (Discrete)

3.2 Binomial & Geometric Distribution

3.2.1 The Binomial Distribution

3.2.2 Calculating Binomial Probabilities

3.2.3 The Geometric Distribution

3.3 Normal Distribution

3.3.1 The Normal Distribution

3.3.2 Standard Normal Distribution

3.3.3 Normal Distribution - Calculations

3.3.4 Finding Sigma and Mu

3.4 Working with Distributions

3.4.1 Modelling with Distributions

3.4.2 Normal Approximation of Binomial

Author: Dan