## Numerical data

Observations that can be measured or counted are known as **numerical data**.

Some examples are:

- the time in minutes to get to school
- the height in centimetres of a 13-year-old girl
- the weight of boys who are in year 9
- the number of people on a tram at any one time.

### Sample means

Rods are known to have lengths between 50 cm and 100 cm. They are measured to the nearest centimetre. A random sample of 20 is taken, the population being all the rods.

The results are:

61 | 79 | 59 | 91 | 72 | 99 | 80 | 73 | 89 | 91 | 54 | 51 |

78 | 56 | 75 | 69 | 90 | 60 | 66 | 89 |

The mean of the sample is 74 cm, correct to the nearest whole number. This provides us with an estimate of the mean lengths of the rods for the population. However, as with categorical data, we need to take more than one sample.

### Variability of sample means

Here are the results from ten more random samples of 20 rods, taken from the same population.

Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Mean | 74 | 77 | 70 | 73 | 74 | 66 | 76 | 70 | 74 | 81 |

The sample means vary from 66 through to 81.

The sampling process was also undertaken with 100 rods. Again, the means are given correct to the nearest centimetre.

Sample number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Mean | 76 | 73 | 77 | 74 | 74 | 74 | 75 | 70 | 75 | 77 |

Here, the sample means vary from 70 through to 77. The sample means vary less with the larger sample size.

### Summary

Care must be taken when using the sample proportion for categorical data, and the sample mean for numerical data, to estimate or predict the proportion or mean of the variable in the population. Both proportions and means vary considerably.