x

### Range

(a measure of spread)

The difference between the highest and lowest elements.

Example:

The set $$\{33,31,42,45,32,13,36,40\}$$ has a highest value of 45 and lowest value of 13 therefore the range is $45-13=32$.

For group data, an estimate of the range can be worked out using $\mathrm{maximum\; possible\; value}-\mathrm{minimum\; possible\; value}$

Example:

Fish Length / cm | Frequency |
---|---|

1 - 10 | 1 |

11 - 20 | 7 |

21 - 30 | 5 |

31 - 40 | 4 |

41 - 50 | 3 |

Here the $\mathrm{minimum\; possible\; value}$ is $1\mathrm{cm}$ and the $\mathrm{maximum\; possible\; value}$ is $50\mathrm{cm}$ so the range is 49.

### Outliers

(comparison between an element and the rest of the set)

An outlier is an extreme data value that doesn't fit the overall pattern.

Example:

The set

$$\{33,31,42,45,32,13,36,40\}$$has a mean of 34 and the range for the data given is 32. One data point to not close to this mean, i.e., 13. If we remove 13 (as it looks like an outlier) the mean is 37 and the range is now 14, so removing what looks like an outlier has little effect on the mean but changes the range considerably. To reject the outlier completely we must look at the context in which the data was collected to confirm why it doesn't fit the same pattern.

### Comparison

(between two sets of data)

To compare two (or more) sets of data, compare the averages (mean, median or mode) of the sets and compare the ranges of the sets.

Example:

Set A

$$\{33,40,52,45,37,50,61,38\}$$has a mean of 44.5 and range of 28.

Set B

$$\{62,70,0,52,7,85,22,96\}$$has a mean of 28 and range of 96.

It can be seen that set A is both higher on average and more consistent (lower range)

Written with Pearson Edexcel Level 1/2 GCSE (9–1) in Mathematics (1MA1) Specification as a guide

### Standard Deviation - a measure of spread:

$${\sigma}^{2}=\frac{\Sigma {({x}_{i}-\stackrel{\_}{x})}^{2}}{n}$$ $${\sigma}^{2}=\frac{\Sigma ({{x}_{i}}^{2}-2{x}_{i}\stackrel{\_}{x}+{\stackrel{\_}{x}}^{2})}{n}$$ $${\sigma}^{2}=\frac{\Sigma {{x}_{i}}^{2}-2\Sigma {x}_{i}\stackrel{\_}{x}+\Sigma {\stackrel{\_}{x}}^{2}}{n}$$ $${\sigma}^{2}=\frac{\Sigma {{x}_{i}}^{2}}{n}-\frac{2\stackrel{\_}{x}\Sigma {x}_{i}}{n}+\frac{n{\stackrel{\_}{x}}^{2}}{n}$$ $${\sigma}^{2}=\stackrel{\_}{{{x}_{i}}^{2}}-2{\stackrel{\_}{x}}^{2}+{\stackrel{\_}{x}}^{2}$$ $${\sigma}^{2}=\stackrel{\_}{{x}^{2}}-2{\stackrel{\_}{x}}^{2}+{\stackrel{\_}{x}}^{2}$$ $${\sigma}^{2}=\stackrel{\_}{{x}^{2}}-{\stackrel{\_}{x}}^{2}$$or "the standard deviation is the mean of the square minus the square of the mean".