The normal distribution
Introduced| English | Meaning |
|---|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ | A continuous, symmetric bell-shaped probability distribution specified by its mean and standard deviation. |
| z-score/zed skɔː/ | A value's distance from the mean measured in standard deviations, calculated as the value minus the mean divided by the standard deviation. |
State the normal assumption
- A normal distribution 正态分布 is a continuous symmetric bell-shaped model, defined by mean and positive standard deviation.
- Those summaries alone do not prove that actual scores, heights or errors are normal. Inspect the data and the proposed modelling assumptions before using normal areas.
How many numbers are needed to describe a normal distribution completely?
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The mean fixes where it sits and the standard deviation fixes how wide it is.
Separate approximate rules from table areas
- Under a normal model, about 68%, 95% and 99.7% fall within one, two and three standard deviations of the mean respectively.
- These rounded empirical-rule values are estimates. With a cumulative table, use its values consistently: $\Phi(2)=0.9772$ gives an upper tail of 0.0228 rather than the rough 0.025.
Under a normal model, approximately what percentage lies within two standard deviations of the mean, using the rounded empirical rule?
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The rounded empirical rule gives about 95%. A cumulative normal table gives a more precise area, so do not treat this rounded value as exact.
Standardise a bound and select its area
- A z-score 标准分 is $z=(x-\mu)/\sigma$. Its sign distinguishes a value below or above the model mean.
- Use $1-\Phi(z)$ for an upper tail and subtract two cumulative values for an interval. Equal z-scores compare relative positions; they do not establish equal test content or learner ability.
A test has mean 62 and standard deviation 8. What is the z-score of 78?
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(78 − 62) ÷ 8 = 2, so the score is two standard deviations above the mean.
A stated normal score model. With mean 62 and standard deviation 8, threshold 78 has $z=(78-62)/8=2$. The approximate empirical rule gives an upper tail near 2.5%; the supplied table gives $P(X>78)=1-0.9772=0.0228$, or 2.28%. Label which method and precision are requested.
For the stated normal model with mean 62 and standard deviation 8, use the rounded 95% empirical rule to estimate the percentage above 78.
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78 is two standard deviations above the mean. The rounded rule leaves 5% outside the central interval, split symmetrically into about 2.5% in each tail. A table gives about 2.28% instead.
Put a normal-distribution question in the order that avoids a tail error.
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The sketch is what tells you whether the answer should be small or large before you trust the arithmetic.
An expected count need not be observed exactly. If a model probability is 0.0228 in 500 trials, expected count is $E=Np=11.4$. Any actual count is an integer and can vary. Sheet 2.8 keeps model probabilities, expected counts and observed findings separate.
The empirical rule assumes a normal model. It is not a rule for every data set with a mean and standard deviation. A strictly positive real quantity may sometimes be approximated by a normal model over a relevant range, but its impossible negative tail still needs consideration.
The empirical rule can be applied to any data set.
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It assumes a normal shape. Income and other skewed data break it, and the answer looks confident and is wrong.