#statistics 

**Measures of variability** - descriptions of the amount by which scores are dispersed or scattered in a distribution

## Usage in determining statistical stability
The smaller the variabilities within groups in experiments are, the greater the statistical stability for the observed mean difference is, thus it is more likely that the observed [[Measures of central tendency|mean]] difference is real and not merely transitory
![[Pasted image 20230827124133.png]]
The OMD in experiment B is more likely to be real than in experiment C.

## Types for quantitative data
### Range
The **range** is the difference between the largest and smallest scores in a sample
Caveats: 1) the value depends on only two scores and fails to use the other; 2) the value of the range tends to increase with increases in the total number of scores

### Interquartile range
**IQR** - the range for the middle 50 percent of the scores (the distance between the 75th percentile and the 25th percentile).
The IQR is not sensitive to the distorting effect of extreme scores, or [[Outliers|outliers]].

### Variance
**Variance** is the mean of all squared deviation scores (deviation score - the distance of a score from the mean)
Same concept as in [[Скорость и давление молекул газа#^a08cdf|среднеквадратичная скорость]] - the scores are squared to eliminate negatives

For population:
$$\LARGE \sigma^2=\frac{SS}{N}$$
For sample:
$$\LARGE s^2=\frac{SS}{n-1}$$

### Standard deviation

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**Standard deviation** is the square root of variance - it is a rough measure of the average (standard) amount by which scores deviate on either side of their mean.

Although the standard deviation usually exceeds the mean absolute deviation, it is nevertheless reasonable to describe the SD as the average amount by which scores deviate on either side of their mean.

For most frequency distribution, a majority of all scores are within one standard deviation on either side of the mean, while only a small minotiry of all scores deviate more than 2 standard deviations on either side of the mean.

For population:
$$\LARGE \sigma=\sqrt{\frac{SS}{N}}$$
For sample:
$$\LARGE s=\sqrt{\frac{SS}{n-1}}$$
In samples, $\LARGE n-1$ is used because of the tendency of $\LARGE n$ estimates to understimate variability in the population because $\LARGE n$ is too large (in generalisations from samples to populations)

Standard formula with [[Degrees of freedom|degrees of freedom]]:
$$\LARGE \sigma (or \space s) = \sqrt{\frac{SS}{\mathrm{d}f}}$$
#### Explanation
![[Pasted image 20230827133519.png]]
![[Pasted image 20230827133536.png]]
![[Pasted image 20230827133546.png]]
![[Pasted image 20230827133600.png]]


### Sum of Squares (SS)
**The sum of squared deviation scores** needed to calculate the variance.

For [[Populations and samples|populations]]:

Definition formula:
$$\LARGE SS = \sum (X - \mu)^2$$,where $\LARGE X-\mu$ - individual deviation score

Computation formula:
$$\LARGE SS=\sum X^2-\frac{(\sum X)^2}{N}$$
For samples:

Def formula:
$$\LARGE SS=\sum (X -\overline X)^2$$
Comp formula:
$$\LARGE SS=\sum X^2-\frac{(\sum X)^2}{n}$$
## Types for qualitative and ranked data
For qualitative and ranked - virtually non-existent, only (evenly divided - max variability, unevenly divided - intermediate variability, concentrated mostly in one class - min variability)