Statistics recap

A working set of notes for applied statistics — descriptive stats, distributions and the everyday tables. Numbers in formulas use μ, σ for a whole population and x̄, s for a sample.

Sampling

Before any statistic means anything, the sample has to represent the population. A good sample is spread across the whole population; a bad sample clusters in one place and misses the rest.

Centre & spread

Two questions about any dataset: where is the middle (central tendency), and how far do values wander from it (spread)?

Central tendency

• Mean — the average. Uses every value, so it gets dragged toward extreme values.
• Median — the middle value when sorted. More resistant to outliers than the mean.
• Mode — the most frequent value. Comes in three flavours:

Spread — variance & standard deviation

Variance is the average squared distance from the mean. Standard deviation is its square root, putting spread back into the original units — roughly “how far is a typical value from the mean?”

A sample divides by n − 1 (Bessel’s correction): a sample tends to underestimate the true spread, so we shrink the denominator to compensate.

Spread — the interquartile range (IQR)

The quartiles cut the sorted data into four. IQR = Q3 − Q1 — the range of the middle 50%.

• Small IQR → values clustered near the centre.
• Large IQR → values spread out.

The distribution curve

One graph for most of it. Move the mean to slide the curve, change the spread to make it taller/narrower or shorter/wider, and add skew to pull a tail. Toggle the layers to see the empirical rule, Chebyshev’s bounds, the mean/median/mode split, and a z-score.

Normal distribution & the empirical rule

A normal distribution is the symmetric bell. Because its shape is fixed and predictable, set proportions fall within each band of the mean:

The smooth bell is a density curve: the y-axis is probability density, not count, and the total area under it is 1. Probability = area under a stretch of the curve.

z-scores

A z-score restates a value as “how many standard deviations from the mean”, so values from different distributions become comparable.

Look up the area (probability) for a z-score at ztable.io.

Chebyshev’s theorem — any shape

The empirical rule needs a normal shape. When the distribution is skewed or unknown, Chebyshev still gives a guaranteed minimum proportion within k standard deviations — it works for any shape, so it’s more conservative.

k must be > 1 (at k = 1 the formula gives 0, so no useful bound). k is just a z-score: k = (x − μ) / σ.

Shape: symmetry, skew & outliers

Where the mean, median and mode sit relative to each other tells you the shape. The mean chases the tail.

▮ mean   ┄ median   ··· mode

Which measures to use

When a tail or outlier would drag the mean and standard deviation around, switch to the median and IQR — both ignore the extremes.

Spotting outliers — the 1.5×IQR rule

A value is an outlier if it falls outside the fences:

Plots

Box & whiskers

Draws the five-number summary (min, Q1, median, Q3, max). The box is the IQR; whiskers reach the last value inside the fences; dots beyond them are outliers. Great for comparing spread and skew, and for seeing outliers at a glance.

Stem & leaf

Keeps the raw values while showing the shape (like a histogram on its side). The stem is the leading digit(s), each leaf the last digit. Best for small datasets. Below: 23, 25, 31, 34, 34, 42, 47, 48, 51.

stem | leaf
  2  | 3 5
  3  | 1 4 4
  4  | 2 7 8
  5  | 1

Histogram, frequency polygon & density curve

A histogram bins continuous data and draws a bar per bin (bars touch). A frequency polygon joins the bar midpoints with a line. A density curve is the smooth, idealised version — the population shape the data is hinting at. All three show centre, spread, skew and modality.

Tables & distributions

A one-way table counts one variable. A two-way (contingency) table cross-tabulates two. From one two-way table you can read every distribution below. Example: 100 students, by year and whether they drink coffee.

• Relative frequency — counts ÷ total, i.e. proportions instead of raw counts.
• Joint distribution — each inner cell ÷ grand total. P(Junior and Yes) = 30/100 = 0.30.
• Marginal distribution — a total (margin) ÷ grand total. P(Junior) = 50/100 = 0.50; P(Yes) = 75/100 = 0.75.
• Conditional distribution — a cell ÷ its row/column total (you’re given one variable). P(Yes given Senior) = 45/50 = 0.90.

The joint distribution as area — each of the 100 squares is one student, grouped by the four cells. The block sizes are the joint proportions: