The measures of central tendency are processes for determining what the central value in a dataset is. The most common is the arithmetic average, or mean – so this value has come to be known as simply the average.

The three measures of central tendency are mean, median and mode.

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## Mean

To calculate the mean (also known as the arithmetic mean or arithmetic average), you take all of the scores, add up their values and divide them by the number you have. Let’s look at the following values of student values out of 10:

4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 |

6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 |

7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |

7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

8 | 8 | 9 | 9 | 9 | 9 | 9 | 10 |

There are 40 values here. If we add them all up, we get the total 280. Dividing by the number of values, we get an average of:

- / 40 = 7

**Median**

The mean is a very common distribution but can be affected by extreme sources. If any values are very high or very low compared to the majority, the mean can be affected. In situations like this, we use the median. The median is the middle value in the set of scores.

For instance, let’s look at a limited set of numbers from the above data set:

4 | 4 | 4 | 5 | 5 | 5 | 6 |

There are 7 values here, so the middle value, 5 becomes the median. In a situation like our full chart above where we have 40 values, we instead have two middle values.

17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |

Taking values 20 and 21 (7 + 7) and dividing them by 2 gives us the median 7.

**Mode**

Finally, the mode is simply the most common score occurring in a distribution. In the full data set above, we have the following values and frequencies:

Value | Frequency |

4 | 3 |

5 | 3 |

6 | 7 |

7 | 12 |

8 | 9 |

9 | 5 |

10 | 2 |

In this case, 7 appears twelve times, so it becomes our median.

**Choosing a Measure of Central Tendency**

The mean is most commonly used – it is the best for symmetric distributions (distributions without major outliers.) The median is best for a skewed distribution or one with outlier(s), while the mode is used in 3 cases:

- One particular score dominates a distribution
- Distribution is bimodal or multimodal
- Data are nominal

**Weighted Mean**

One special case of the mean is the “weighted mean”, where some values are “weighted” or contribute more to the total value than others. The data set from above is presented here:

Value | Frequency |

4 | 3 |

5 | 3 |

6 | 7 |

7 | 12 |

8 | 9 |

9 | 5 |

10 | 2 |

To calculate the weighted mean, we multiply each value by its frequency, before dividing by the frequency. This is similar to the mean as you’ll see:

- 3×4 + 3×5 + 7×6 + 12×7 + 9×8 + 5×9 + 10×2

= 12 + 15 + 42 + 84 + 72 + 45 + 20

= 290 - We divide by the original frequencies:

3 + 3 + 7 + 12 + 9 + 5 + 2

= 41

- And now we’ll divide the top by the bottom:290 / 41 = 7.073

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