Tool

Binomial to normal

Press Play and n steps up from 5 to 200. Each binomial is drawn on the standardised scale, (k − np)/σ, with its bars' areas kept, so the standard normal curve stays where it is while the bars grow thinner and settle onto it. The question is the chance of at most k successes, with k a fixed share of n; the readout compares the exact answer with the normal's, with and without the continuity correction.

Goes with chapter 8. The standard normal and z-scores

Set-up

p = 0.3

Probability of success p

k = 0.25 × n, rounded down

k as a share of n

The question: P(X ≤ k), with k = 0.25 × n rounded down, for each n.

Playback speed

Bars counted by P(X ≤ k)
The other bars
The standard normal curve, shaded to k + ½

At n = 200: P(X ≤ 50) is exactly 0.0695; the normal gives 0.0713 with the continuity correction and 0.0614 without it.

Every step in numbers

n	k	Exact	Normal, corrected	Normal, not corrected	Error, corrected	Error, not corrected	np ≥ 5, n(1 − p) ≥ 5	Explorer
5	1	0.5282	0.5000	0.3128	−0.0282	−0.2154	no	Explorer
10	2	0.3828	0.3650	0.2451	−0.0177	−0.1377	no	Explorer
15	3	0.2969	0.2866	0.1990	−0.0103	−0.0979	no	Explorer
20	5	0.4164	0.4036	0.3128	−0.0127	−0.1036	yes	Explorer
25	6	0.3407	0.3313	0.2563	−0.0094	−0.0843	yes	Explorer
30	7	0.2814	0.2750	0.2128	−0.0063	−0.0686	yes	Explorer
40	10	0.3087	0.3024	0.2451	−0.0064	−0.0637	yes	Explorer
50	12	0.2229	0.2202	0.1773	−0.0027	−0.0456	yes	Explorer
60	15	0.2438	0.2406	0.1990	−0.0032	−0.0448	yes	Explorer
80	20	0.1978	0.1966	0.1646	−0.0013	−0.0333	yes	Explorer
100	25	0.1631	0.1631	0.1376	−0.0001	−0.0255	yes	Explorer
120	30	0.1359	0.1366	0.1160	+0.0007	−0.0199	yes	Explorer
150	37	0.0890	0.0907	0.0770	+0.0017	−0.0120	yes	Explorer
200	50	0.0695	0.0713	0.0614	+0.0018	−0.0081	yes	Explorer

What to look for

At n = 5 the bars are wide and the curve is a rough guide; by n = 200 the bars are thin slivers that follow the curve closely. The error of the corrected normal shrinks steadily as n grows. Without the correction the error shrinks more slowly, because it always leaves out half of the bar at k.

Set p to 0.05 and watch the early steps: the binomial is badly skewed, np is below 5, and the symmetric curve fits it poorly. That is what the rule of thumb in the table checks. Set p to 0.5 and the fit is good almost at once.