# Summary statistics and visualisation for one variable

## Basis statistics for a single variable

### N

The is the size of the sample.

### Sample range

#### Minimum

This is the smallest value in the sample.

#### Maximum

This is the largest value in the sample.

#### Range

This is the difference between the maximum and minimum.

#### Median

This is the value whereby 50% of the sample can be found below the value.

#### Percentiles

The $$x$$th percentile is the value by which $$x\%$$ of the values can be found below it.

#### Interquartile range

This is the differnence between the $$25$$th percentile and the $$75$$th percentile.

### Sample mode

The is the most common value in the sample.

## Sample moments

### Sample mean

We previously defined the population mean is defined as $$\mu=E[X]$$.

The sample mean is defined as $$\bar x = \dfrac{1}{n}\sum_i x_i$$.

#### Centred mean

We can subtract the mean from each entry in the sample. This will leave a new mean of $$0$$. This is convenient for many calculations.

### Sample variance

We previously defined the population variance as $$\sigma^2=E[(X-\mu)^2]$$.

We define the sample variance as $$\sigma^2=\dfrac{1}{n}\sum_i(x_i-\bar x)^2$$.

We can calculate this using matrices:

$$M=X-\bar x$$

$$\sigma^2=\dfrac{1}{n}M^TM$$.

#### Centred variance

If $$\bar x =0$$ then:

$$\sigma^2=\dfrac{1}{n}X^TX$$.

## Updating statistics

### Updating the mean

$$\bar x_{n+1} = \dfrac{n\bar x_n+x_{n+1}}{n+1}$$

### Updating the variance

If it is centred:

$$\sigma_n^2=\dfrac{1}{n}X_n^TX_n$$

So:

$$\sigma_{n+1}^2=\dfrac{n\sigma_n^2 +x_{n+1}^tx_{n+1}}{n+1}$$