### 1. Descriptive Statistics: Measures of Central Tendency

### Sensitivity to Outliers

An important consideration when comparing the mode, median, and mean as measures of centrality is their *sensitivity to* outliers.

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An **outlier **is an exceptionally high or low score that does not conform to the pattern observed for the majority of the data.

Outliers can be the result of a measurement error or a mistake made during the entry of the data, but more often the score is a legitimate exceptional case.

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A statistical measure is said to be

**sensitive**to outliers when it is influenced by the presence of outliers in the dataset.

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Both the mode and the median are measures of centrality which are *not sensitive* to the presence of outliers in the dataset. The mean, on the other hand, is very *sensitive* to the presence of outliers.

The following example helps illustrate the concept of sensitivity to outliers in the context of central tendency.

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Measures of Centrality and Sensitivity

#\phantom{000000000000}#** Dataset**

Consider the following set of #n=13# scores:

#1,\, 1,\, 2,\, 4,\, 5,\, 5,\, 6,\, 8,\, 8,\, 8,\, 9,\, 10,\, 11#

**Measures of Centrality**

- Mode #= 8#
- Median #= 6#
- Mean #= \dfrac{78}{13} = 6#

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Now, consider what happens when the score #X = 11# is changed into an outlier, such as #X=76#.

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#\phantom{000000000000}#** Dataset**

The new set of #n=13# scores is:

#1,\, 1,\, 2,\, 4,\, 5,\, 5,\, 6,\, 8,\, 8,\, 8,\, 9,\, 10,\, \boldsymbol{76}#

**Measures of Centrality**

- Mode #= 8#
- Median #= 6#
- Mean #= \dfrac{143}{13} = 11#

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Both the mode and the median remain unchanged as the result of changing a score into an outlier:

- The most frequently occurring value in the dataset stays the same, namely #X=8#.
- Likewise, the middlemost score also stays the same, namely #X_7=6#.

This demonstrates that both the mode and the median are examples of measures that *insensitive* to the presence of outliers in the dataset.

The mean, however, changes quite a lot as a result of the outlier. Since every score in the dataset contributes equally to the mean, a single high or low score can have a drastic effect on the mean, especially if the dataset is relatively small.