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 — not for individual kiwifruit, and not a guarantee.
• A wider interval means more uncertainty about where the population median is.

What makes the interval narrower

• A larger sample ($n$ up) → $\sqrt{n}$ up → margin down → narrower interval, so a more precise estimate.
• A smaller IQR (less spread in the data) → smaller margin → narrower interval.
• Narrow intervals are what let you eventually make the call between two groups.