The normal distribution · La loi normale
| English | Français |
|---|---|
| normal distribution/ˈnɔːml ˌdɪstrɪˈbjuːʃn/ | distribution normale |
| z-score/zed skɔː/ | score z |
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? · Combien de nombres sont nécessaires pour décrire complètement une distribution normale ?
The mean fixes where it sits and the standard deviation fixes how wide it is. · La moyenne fixe où elle se situe et l'écart-type fixe sa largeur.
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? · Selon un modèle normal, quel pourcentage se situe approximativement dans deux écarts-types de la moyenne, en utilisant la règle empirique arrondie ?
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. · La règle empirique arrondie donne environ 95 %. Un tableau de la normale cumulative donne une aire plus précise, donc ne traitez pas cette valeur arrondie comme exacte.
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? · Un test a une moyenne de 62 et un écart-type de 8. Quelle est la cote z de 78 ?
(78 − 62) ÷ 8 = 2, so the score is two standard deviations above the mean. · (78 − 62) ÷ 8 = 2, donc la note est de deux écarts-types au-dessus de la moyenne.
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. · Pour le modèle normal stipulé avec une moyenne de 62 et un écart-type de 8, utilisez la règle empirique arrondie de 95 % pour estimer le pourcentage au-dessus de 78.
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. · 78 est deux écarts-types au-dessus de la moyenne. La règle arrondie laisse 5 % hors de l'intervalle central, réparti symétriquement en environ 2,5 % dans chaque queue. Un tableau donne environ 2,28 % au lieu de cela.
Put a normal-distribution question in the order that avoids a tail error. · Placer une question de distribution normale dans l'ordre évitant une erreur de queue.
The sketch is what tells you whether the answer should be small or large before you trust the arithmetic. · L'esquisse indique si la réponse devrait être petite ou grande avant de faire confiance aux calculs.
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. · La règle empirique peut s'appliquer à n'importe quel jeu de données.
It assumes a normal shape. Income and other skewed data break it, and the answer looks confident and is wrong. · Elle suppose une forme normale. Le revenu et autres données asymétriques la contredisent, et la réponse paraît sûre mais est fausse.