Building an informal confidence interval

Why one sample median is not enough

• Your sample median is a single estimate of the population median. Because of sampling variability, the true value is probably near it, not exactly it.
• Instead of one number, you quote a range of believable values for the population median. That range is the informal confidence interval.

The informal confidence interval

• The interval is centred on the sample median and reaches out an equal amount on each side.

• It gives a range in which the population median is believed to lie.

The formula

$$\text{median} \; \pm \; 1.5 \times \dfrac{\text{IQR}}{\sqrt{n}}$$

• median — the sample median (the centre of the interval).
• IQR — the interquartile range, $Q_3 - Q_1$ (the sample's spread).
• $n$ — the sample size (how many values you measured).
• $\sqrt{n}$ — the square root of the sample size. Dividing by it means bigger samples give smaller intervals.
• $1.5$ — a fixed multiplier for this informal method.

Building the interval step by step

• Find the sample median and the IQR ($Q_3 - Q_1$).
• Compute the margin $E = 1.5 \times \dfrac{\text{IQR}}{\sqrt{n}}$.
• Subtract $E$ from the median for the lower end, add $E$ for the upper end.
• Write the interval as $(\text{median} - E, \; \text{median} + E)$, in the units of the context.

What the interval means

• It is a range of believable values for the population median, based on this sample.
• It is not about individual values. The interval $(104, 116)$ does not say most kiwifruit weigh between 104 g and 116 g — plenty weigh less than 98 g or more than 122 g. It is a statement about the median of the whole orchard, which is a single fixed number we are trying to locate.
• It is not a guarantee. The method usually captures the population median, but any one interval built from any one sample may miss it.
• A wider interval means more uncertainty about where the population median sits, not more variation in the fruit.
• Say it in context, in full. "It is a believable range for the median weight of all kiwifruit from Orchard A" — naming the population and the variable is what earns the interpretation marks.

Why the formula has the shape it does

Each part of $1.5 \times \dfrac{\text{IQR}}{\sqrt{n}}$ is doing a job, and understanding them makes the formula hard to misuse.

• IQR on the top — the more spread out the data, the less precisely one sample pins down the centre, so a bigger IQR widens the interval.
• $\sqrt{n}$ on the bottom — a bigger sample gives a more reliable estimate, so more data narrows it.
• The square root, not $n$ itself — this is why precision improves slowly. To halve the interval you must quadruple the sample size, not double it.
• The 1.5 is a fixed multiplier chosen so this informal method behaves roughly like the formal intervals used at Level 3.

What makes the interval narrower

• A larger sample ($n$ up) → $\sqrt{n}$ up → margin down → narrower interval, and a more precise estimate.
• A smaller IQR (less spread in the data) → smaller margin → narrower interval.
• You control one of these and not the other. Sample size is your decision; the spread of the population is a fact about the world, and pretending otherwise by discarding awkward values is not an option.
• Narrow intervals are what let you eventually make the call between two groups — if the two intervals do not overlap, the difference is worth claiming.