# Summarize Numerical Data

## Measure Centre

Mean, median, and mode are three kind presentation of the “centre”.

### Mean

**Mean**: is the “average” you’re used to, where you add up all the numbers and then divide by the number of numbers.

### Ｍedian

**Median** is the “middle” value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first.

### Mode

**Mode** is the value that occurs most often. If no number is repeated, then there is no mode for the list.

## An example

Find the mean, median, mode, and range for the following list of values:

**13, 18, 13, 14, 13, 16, 14, 21, 13**

Mean: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

Median: 13, 13, 13, 13, **14**, 14, 16, 18, 21

Mode: the occur of 13 is more than other numbers. So 13 is the mode.

**mean:****15****median:****14****mode:****13**

## Measure Spread

**Interquartile range** (IQR), **Range and Standard Deviation** are two summarise of data spread.

**Interquartile range** (IQR)

The **interquartile range** (IQR) is a measure of variability, based on dividing a data set into **quartiles**.

IQR also relies on Q1, Q2, and Q3, respectively.

- Q1 is the “middle” value in the
*first*half of the rank-ordered data set. - Q2 is the
**median**value in the set. - Q3 is the “middle” value in the
*second*half of the rank-ordered data set.

**IQR=Q3-Q1**

### Range

**Range** is the difference between the largest value and smallest value.

### Standard Deviation

The Standard Deviation is a measure of how spread out numbers are. Its symbol is **σ** (the greek letter sigma)

The formula is easy: it is the** square root** of the **Variance. **

**Population** Standard Deviation:

**Sample** Standard Deviation:

## Variance

The Variance is defined as: the average of the squared differences from the Mean.

## An example

**5, 8, 4, 4, 6, 3, 8**

Put them in **order:** 3, 4, 4, 5, 6, 8, 8

Cut the list into quarters:

- Quartile 1 (Q1) =
**4** - Quartile 2 (Q2), which is also the Median, =
**5** - Quartile 3 (Q3) =
**8** - IQR= Q3-Q1= 8-4 =
**4** - Range = 8 -3 =
**5** - Standard Deviation =
**1.988**