Statistics: Mean, Median, Standard Deviation & Z-Scores

\bar{x} = \dfrac{\sum x_i}{n}

Sort the data. With n odd, it is the middle value. With n even, it is the average of the two middle values.

The most frequent value. A data set can have no mode, one mode, or several.

5

5.5

The median. One extreme value can drag the mean a long way, but barely moves the median.

Right-skewed (long tail to the right): mean is usually greater than the median. Left-skewed: mean is usually less than the median.

\text{max} - \text{min}

Sample: s^2 = \dfrac{\sum (x_i - \bar{x})^2}{n - 1} Population: \sigma^2 = \dfrac{\sum (x_i - \mu)^2}{N}

Bessel's correction. Deviations are measured from the sample mean, which makes them slightly too small; dividing by n - 1 makes s² an unbiased estimate of σ².

The square root of the variance. It is in the same units as the data, so it reads as a typical distance from the mean.

\text{IQR} = Q_3 - Q_1 , the spread of the middle 50% of the data

A value is an outlier if it is below Q_1 - 1.5\,\text{IQR} or above Q_3 + 1.5\,\text{IQR} .

Minimum, Q1, median, Q3, maximum. A box plot draws exactly these five numbers (with outliers plotted separately).

z = \dfrac{x - \mu}{\sigma}

1.5

Only applies to roughly bell-shaped, symmetric distributions.

CV = \dfrac{s}{\bar{x}} \times 100\%

A parameter describes a whole population (μ, σ, p). A statistic describes a sample (x̄, s, p̂) and is used to estimate the parameter.

Adding c: mean shifts by c, SD is unchanged. Multiplying by k: mean and SD are both multiplied by |k| (the mean by k).

Categorical: groups or labels (eye color, major). Quantitative: numbers you can do arithmetic on (height, income). Quantitative variables are discrete (counts) or continuous (measurements).

The strength and direction of a linear relationship, from -1 to 1 . r = 0 means no linear relationship (there may still be a curved one).

Quantitative: histogram, dot plot, box plot, stem-and-leaf. Categorical: bar chart, pie chart. Two quantitative variables: scatterplot.