The sampling distribution of a sample proportion describes how the sample proportion p̂ varies from one random sample to the next. Its mean is the population proportion p, and its standard deviation is √(p(1 − p)/n). When np and n(1 − p) are both at least 10, its shape is approximately normal.
The central limit theorem (CLT) says that when you take large random samples from any population, the sampling distribution of the sample mean is close to normal, whatever the population’s shape. Its mean is μ and its standard deviation is σ/√n. A common rule of thumb is n ≥ 30.
A sampling distribution is the distribution of a statistic, like a sample mean or a sample proportion, over all possible random samples of the same size from a population. It shows how much the statistic changes from sample to sample, and whether it is centered on the true value for the population.
The normal approximation to the binomial uses a normal curve to estimate binomial probabilities when the number of trials is large. If np ≥ 10 and n(1 − p) ≥ 10, the binomial count X is close to normal, with mean np and standard deviation √(np(1 − p)).
A normal probability plot is a graph that shows whether a data set is close to normal. Each data value is plotted against the z-score it would have in a perfectly normal data set of the same size. If the points fall close to a straight line, the data are close to normal.
Normal probability calculations answer two questions about a normal distribution: what is the probability that a value falls in a range, and what value cuts off a given percent? For a normal distribution probability, change each value to a z-score and use a z-table or a calculator. For a cutoff, work backward.
The standard normal distribution is the normal distribution with mean 0 and standard deviation 1, written N(0, 1). Its values are z-scores. A z-table or a calculator gives the area to the left of any z. From that, you can find the area to the right of z or between two z-scores.
The empirical rule, also called the 68-95-99.7 rule, says that in a normal distribution about 68% of values fall within 1 standard deviation of the mean, about 95% fall within 2, and about 99.7% fall within 3. It lets you estimate percents and probabilities without a table.
A normal distribution is a symmetric, bell-shaped distribution of a continuous variable. Two numbers set it: the mean μ, which is the center of the curve, and the standard deviation σ, which sets how spread out it is. Many real measurements, like heights and birth weights, are close to normal.
A density curve is a curve that describes the distribution of a continuous variable. It always lies on or above the horizontal axis, and the total area under it is exactly 1. The area under the curve over an interval is the proportion of values, or the probability, in that interval.