# Statistics » Mode, median and mean

## Contents

ModeMedian

Mean

Or were you looking for mean deviation from the mean?

## Measures of central tendancy

The mode, median and mean are called measures of central tendancy.

## Mode

The value that is most common in a row of data is the mode. In other words, it is the value with the highest frequency.

With a distribution in classes is the class with the highest frequency the *modal class*.

If there are two values with the highest frequency, there is no mode.

*Example 1*

Given is the row: 1, 2, 2, 3, 3, 4, 5, 6, 6, 6, 6, 7

The 6 is most common. So 6 is the mode.

*Example 2*

mark | frequency |

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

The mark 7 appears the most times (six times). The mode is 7.

*Example 3*

class | frequency | |

40 to 50 50 to 60 60 to 70 70 to 80 80 to 90 |
3 4 4 5 2 |
<= Modal class |

## Median

The median is the middle value in a row of data in ascending order.

The median divides the data in two, 50% of the values is lower than the median and 50% higher than the median.

### Odd number of values:

Take the middle number.

### Even number of values:

There is no middle number. There are two numbers who make up the middle. The mean of those two numbers is the median.

### How do you calculate which value is the middle value?

`number of the middle value` = `total number of values` + 12

*Example 1*

What is the median of 1, 6, 4, 3, 2, 8, 7, 6, 12 and 3?

First put them in ascending order: 1, 2, 3, 3, 4, 6, 6, 7, 8, 12.

There are 10 values, so the middle value is the 10 + 12 = 5.5th number.

Therefore you need the fifth and sixth value.

Those values are 4 and 6.

The median is 4 + 62 = 5.

*Example 2*

Given is the following frequency table. Calculate the median.

mark | frequency |

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

There is a total of 3 + 4 + 5 + 6 + 2 + 4 = 24 values.

The middle value is the 24 + 12 = 12.5th number.

So you need to find the 12th and 13th number.

Start counting from the top, using the frequencies.

The 12th number is a 6. The 13th number is a 7.

The median is therefore 6 + 72 = 6.5.

### Quartiles

The median of the first half of the numbers is called the lower quartile and the median of the second half of the numbers is called the upper quartile.

## Mean

The mean can be calculated in the following way:

`mean` = `sum of the values``total number of values`

*Means for the examples above*

Example 1: 1 + 6 + 4 + 3 + 2 + 8 + 7 + 6 + 12 + 310 = 5210 = 5.2

Example 2: 4×3 + 5×4 + 6×5 + 7×6 + 8×2 + 9×43 + 4 + 5 + 6 + 2 + 4 = 15624 = 6.5

### Mean of a distribution in classes

What if you have a distribution in classes?

In that case you can only make an *estimation* of the mean.

1. Calculate the midpoint of each class.

2. Multiply the midpoint with the frequency and add the outcomes.

3. Divide the sum of the outcomes by the sum of the frequencies.

*Example*

length in cm |
freq. |
midpoint |
calculation |
||||

150 to 160 | 3 | 155 | 3 × 155 = | 465 | |||

160 to 170 | 4 | 165 | 4 × 165 = | 660 | |||

170 to 180 | 5 | 175 | 5 × 175 = | 875 | |||

180 to 190 | 4 | 185 | 4 × 185 = | 740 | |||

190 to 200 | 2 | + | 195 | 2 × 195 = | 390 | + | |

1 | 8 | 3130 |

Estimation of the mean is 3130 : 18 ≈ 173.9 cm.