Percentile Rank

The percentile rank of a score is the percentage of scores in the distribution that are equal to or lower than it.

Percentile

Percentiles are the values that divide a distribution of scores into one hundred equal parts.
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The $P^{th}$ percentile of distribution is the value such that $P$ percent of scores are equal to or below it.

Quartile

Quartiles are the values that divide a distribution of scores into four equal parts.
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The first $(Q_1)$, second $(Q_2)$, and third $(Q_3)$ quartiles are equal to the $25^{th}$, $50^{th}$, and $75^{th}$ percentile, respectively.

Mode

The mode of a distribution is the most frequently observed value.

Median

The median is the midpoint of a distribution. It is the score that divides a distribution into two equal halves.
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The median is equivalent to the $50^{th}$ percentile, $P_{50}$, meaning exactly $50\%$ of the scores in a distribution are equal to or less than the median (see 'Percentiles').

Mean

For a sample of $n$ observed values of variable $X$, the sample mean equals:

$$\bar{X}=\cfrac{\sum{X}}{n}$$

Range

The range of a distribution is the difference between the minimum and maximum score:

$$\text{Range}=X_{max}-X_{min}$$

Interquartile Range

The interquartile range (IQR) of a distribution is the difference between the third and first quartile of a distribution (see 'Quartiles'):

$$\text{IQR}=Q_{3}-Q_{1}$$

Variance

The variance of a distribution is the average of the squared deviation scores.
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To calculate the sample variance, make use of the following formula:

$$s^2 = \cfrac{\sum{(X-\bar{X})^2}}{n-1}$$

Standard deviation

The standard deviation of a distribution is the positive square root of the variance.
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To calculate the sample standard deviation, make use of the following formula:

$$s = \sqrt{\cfrac{\sum{(X-\bar{X})^2}}{n-1}}\phantom{000}\text{or}\phantom{000}\sqrt{\cfrac{\sum{X^2}-\cfrac{(\sum{X})^2}{n}}{n-1}}$$

Z-scores

A $\boldsymbol{Z}$-score measures a score's distance from the mean in terms of standard deviation units (see 'Standard deviation').
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To turn a sample score $X$ into a $Z$-score, make use of the following formula:

$$Z = \cfrac{X - \bar{X}}{s}$$

Covariance

The covariance measures the direction of the linear relationship between two quantitative variables.
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To calculate the sample covariance between two variables $X$ and $Y$, make use of the following formula:

$$s_{\small{X,Y}}=\cfrac{\sum{\bigg((X-\bar{X})(Y-\bar{Y})\bigg)}}{n-1}$$

Pearson Correlation Coefficient

The Pearson Correlation Coefficient is the standardized form of the covariance (see 'Covariance').
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It is used to measure the direction and strength of the linear relationship between two quantitative variables.
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To calculate the sample Pearson Correlation Coefficient between two variables $X$ and $Y$, make use of the following formula:

$$r_{\small{X,Y}}=\cfrac{s_{\small{X,Y}}}{s_{\small{X}} s_{\small{Y}}}=\cfrac{1}{n-1}\sum{\Bigg(\bigg(\cfrac{X - \bar{X}}{s_{\small{X}}} \bigg)\bigg(\cfrac{Y - \bar{Y}}{s_{\small{Y}}} \bigg)\Bigg)}$$