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Population Distribution — Definition, Formula & Examples

Population distribution is the pattern of values a variable takes across every individual in an entire population, not just a sample. It tells you which values occur and how often each one occurs in the whole group.

For a variable XX measured on a population of NN individuals, the population distribution assigns to each possible value (or interval of values) the proportion of the NN individuals that have that value. Numerical summaries of this distribution, such as the population mean μ\mu and population standard deviation σ\sigma, are called parameters. Parameters are fixed constants, but in practice their values are usually unknown.

Key Formula

μ=∑xiNσ=∑(xi−μ)2N\mu = \frac{\sum x_i}{N} \qquad \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
Where:
  • μ\mu = population mean (center of the population distribution)
  • σ\sigma = population standard deviation (spread of the population distribution)
  • xix_i = the value of the variable for individual i
  • NN = the number of individuals in the entire population

How It Works

You describe a population distribution the same way you describe any distribution: by its shape, center, spread, and any unusual features. When you can measure every individual, as in a census, you can build it directly with a relative frequency table or histogram. Most real populations are too large to measure, so you take a random sample and use the sample distribution to estimate the population distribution. Statisticians also model population distributions with smooth density curves, such as a normal curve for adult heights.

Worked Example

Problem: A small company has exactly 10 employees, and this is the entire population of interest. The number of children each employee has: 1 employee has 0, 2 have 1, 4 have 2, 2 have 3, and 1 has 4. Describe the population distribution and find its parameters μ\mu and σ\sigma.
Step 1: Convert each count to a proportion of the population by dividing by N=10N = 10. The relative frequencies for 0, 1, 2, 3, 4 children are 0.1, 0.2, 0.4, 0.2, 0.1. They sum to 1. The distribution is symmetric with a single peak at 2.
410=0.4 for x=2\frac{4}{10} = 0.4 \text{ for } x = 2
Step 2: Find the population mean. Add every individual's value and divide by NN.
μ=0(1)+1(2)+2(4)+3(2)+4(1)10=2010=2\mu = \frac{0(1) + 1(2) + 2(4) + 3(2) + 4(1)}{10} = \frac{20}{10} = 2
Step 3: Find the population standard deviation. Divide by NN, not N−1N - 1, because you have the whole population.
σ=1(0−2)2+2(1−2)2+4(2−2)2+2(3−2)2+1(4−2)210=1210≈1.10\sigma = \sqrt{\frac{1(0-2)^2 + 2(1-2)^2 + 4(2-2)^2 + 2(3-2)^2 + 1(4-2)^2}{10}} = \sqrt{\frac{12}{10}} \approx 1.10
Answer: The population distribution is symmetric and unimodal, centered at μ=2\mu = 2 children, with spread σ≈1.10\sigma \approx 1.10 children. These are parameters because they describe all 10 employees.

Visualization

Why It Matters

In AP Statistics Units 5 through 7, the shape of the population distribution decides which inference methods are valid. If the population distribution is normal, the sampling distribution of xˉ\bar{x} is exactly normal for any sample size. If the population is skewed, you need a large sample (commonly n≥30n \ge 30) before the Central Limit Theorem lets you treat xˉ\bar{x} as approximately normal.

Common Mistakes

Mistake: Treating the distribution of one sample as if it were the population distribution.
Correction: A sample distribution only estimates the population distribution and varies from sample to sample. Use xˉ\bar{x} and ss for sample statistics, and save μ\mu and σ\sigma for population parameters.
Mistake: Dividing by N−1N - 1 when computing the standard deviation of an entire population.
Correction: The n−1n - 1 divisor corrects for estimating from a sample. When you have data on every individual, divide by NN to get σ\sigma.

Related Terms