# 1.6. Resolutions for the situation or issue are supported by the data and are validated in terms of the context.

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**Range**

The range of a sample (or a data set) is a measure of the spread or the dispersion of the observations. It is the difference between the largest and the smallest observed value of some quantitative characteristic and is very easy to calculate. A great deal of information is ignored when computing the range since only the largest and the smallest data values are considered; the remaining data is ignored. The range value of a data set is greatly influenced by the presence of just one unusually large or small value in the sample (outlier).

*Examples*

- The range of 65,73,89,56,73,52,47 is 89-47 = 42.
- If the highest score in a 1st year statistics exam was 98 and the lowest 48, then the range would be 98-48 = 50.

**Inter-Quartile Range (IQR)**

The inter-quartile range is a measure of the spread of or dispersion within a data set. It is calculated by taking the difference between the upper and the lower quartiles. For example:

Data | 2 3 4 5 6 6 6 7 7 8 9 | |

Upper quartile | 7 | |

Lower quartile | 4 | |

IQR | 7 – 4 = 3 |

The IQR is the width of an interval that contains the middle 50% of the sample, so it is smaller than the range and its value is less affected by outliers.

**Quantile**

Quantiles are a set of ‘cut points’ that divide a sample of data into groups containing (as far as possible) equal numbers of observations.

**Percentile**

Percentiles are values that divide a sample of data into one hundred groups containing (as far as possible) equal numbers of observations. For example, 30% of the data values lie below the 30th percentile.

**Quartile**

Quartiles are values that divide a sample of data into four groups containing (as far as possible) equal numbers of observations. A data set has three quartiles. References to quartiles often relate to just the outer two, the upper and the lower quartiles; the second quartile being equal to the median. The lower quartile is the data value a quarter way up through the ordered data set; the upper quartile is the data value a quarter way down through the ordered data set.

*Example*:

Data | 6 47 49 15 43 41 7 39 43 41 36 | |

Ordered Data | 6 7 15 36 39 41 41 43 43 47 49 | |

Median | 41 | |

Upper quartile | 43 | |

Lower quartile | 15 |

**Sample Variance**

Sample variance is a measure of the spread of or dispersion within a set of sample data.

The sample variance is the sum of the squared deviations from their average divided by one less than the number of observations in the data set. For example, for n observations x1, x2, x3, … , xn with sample mean.

**Standard Deviation**

Standard deviation is a measure of the spread or dispersion of a set of data.

It is calculated by taking the square root of the variance and is symbolised by s.d, or s. In other words.