# Topic 2 Measures of central tendency and dispersion

## 2.1 Central tendency

- Mean
- Median
- Mode

**Example**

We asked 10 people to report the number of books they each read last month.

They reported the following:

5, 3, 4, 3, 0, 1, 4, 4, 8, 0.

**Some notation**

Before we calculate the mode we need to know the following notation (this will also be useful for calculating the mean, variance, standard deviation…)

- \(x\): refers to each value taken by a given variable in the sample. In this example, the variable “number of books” takes the values: 0, 1, 3, 4, 5 and 8.
- \(f\): refers to the
*frequency*of each of the \(x\) values (the number of times individuals reported this value)

Easy way to calculte: write the following table:

Values in order: 0, 0, 1, 3, 3, 4, 4, 4, 5, 8.

x | f | fx |
---|---|---|

0 | 2 | 0 |

1 | 1 | 1 |

3 | 2 | 6 |

4 | 3 | 12 |

5 | 1 | 5 |

8 | 1 | 8 |

\(\sum x = 21\) | \(\sum f=10\) | \(\sum fx=32\) |

### 2.1.1 Mean

The mean is the unweighted average of the values in the sample.

\[\overline{x} = \frac{1}{n}\sum x_i\] Note tha \(x_i\) refers to each observation reported in the sample (it is not the same as \(x\)).

This can take a long time to calculate by hand.

Easier way to calculate the mean (see Prof. Hassad’s notes):

\[\overline{x} = \frac{\sum fx}{\sum f} = \frac{32}{10} = 3.2\]

### 2.1.2 Median

**Step 1:** list all values in ascending order.

0, 0, 1, 3, 3, 4, 4, 4, 5, 8.

**Step 2:** Do I have an odd or even total number of reported values?

*Even total*: There are two middle values. The median is the average between those two values.

In the number of books example, the median is: \[ Median = \frac{3+4}{2} = 3.5\]

*Odd total:* There is only one middle value. The median is this middle value.

### 2.1.3 Mode

The mode is the value with highest frequency (highest f).

How about the following:

x | f |
---|---|

0 | 2 |

1 | 3 |

3 | 3 |

4 | 3 |

5 | 1 |

8 | 1 |

All of those are the mode.

Intuition: How many winners are there if runners tie in first place in a given competition?

How about the following?

x | f |
---|---|

0 | 1 |

1 | 1 |

3 | 1 |

4 | 1 |

5 | 1 |

8 | 1 |

Here there is no mode.

**Normal distribution:**

Mean = Median = Mode

Mode, Mean, and Meadian are useful to discribe the shape of a distribution.

But they are not enough! We need new tools.

## 2.2 Measures of dispersion

- Variance
- Standard deviation
- Range
- Skewness

### 2.2.1 Standard deviation

Use the same table and expand it:

Note that \(\overline{x}\) is the mean you just calculated.

\(x\) | f | fx | \(x - \overline{x}\) | \((x - \overline{x})^2\) | \(f(x - \overline{x})^2\) |
---|---|---|---|---|---|

… | … | … | … | … | … |

. | \(\sum f\) | \(\sum fx\) | \(\sum above = A\) |

Calculate the standard deviation:

Or, more formally:

\[sd = \sqrt{\frac{f(x - \overline{x})^2}{n-1} }\]

\[sd = \sqrt{\frac{A}{n-1} }\]

Note that the standard deviation of a standard normal distribution is 1.

### 2.2.2 Variance:

\[variance = sd^2\]

Note that the variance of a standard normal distribution is 1.

### 2.2.3 Range

Range = Max-Min

### 2.2.4 Skewness

- If skewness is more than 1 or less than -1, then we have a highly skewed distribution.
- If skewness is between -1 and -1/2 or between 1 and 1/2, then we have a moderatly skewed distribution.
- If skewness is between -1/2 and 1/2, then we have an approximatly symmetric distribution (or a negligible degree of skewness).

### 2.2.5 Example of calculations

The following are data for the variable: number of movies watched in the past month.

1; 5; 5; 4; 5; 4; 0; 15; 1; 0; 0; 1.

Calculate: mean, media, mode, variance, standard deviation and range.

\(x\) | f | fx | \(x - \overline{x}\) | \((x - \overline{x})^2\) | \(f(x - \overline{x})^2\) |
---|---|---|---|---|---|

0 | 3 | 0 | -3.4 | 11.56 | 34.68 |

1 | 3 | 3 | -2.4 | 5.76 | 17.28 |

4 | 2 | 8 | 0.6 | 0.36 | 0.72 |

5 | 3 | 15 | 1.6 | 2.56 | 7.68 |

15 | 1 | 15 | 11.6 | 134.56 | 134.56 |

. | \(\sum f = 12\) | \(\sum fx = 41\) | \(\sum above = 194.92\) |

\[\overline{x} = \frac{\sum fx}{\sum f} = \frac{41}{12} = 3.4\]

Mode = 0 , 1 , 5

Median = (4 + 1) / 2 = 2.5

\[sd = \sqrt{\frac{194.92}{11} } = \sqrt{17.72} = 4.2\] \[sd^2 = 17.72\]

\[Range = 15 - 0 = -15\]