The mean, usually called the average, is a quick way to describe the central location of numerical data. Add every observed value, then divide by how many values there are. Instead of inspecting thousands of records one by one, a mean gives one number that can stand in for where the data tends to sit. section 2.12 section 2.13
Picture each value resting on a number-line seesaw with equal weight. The mean is the fulcrum: the point that would balance the seesaw. That makes it a useful representative value, whether the numbers are dog weights, sale amounts, or survey responses. section 2.13 section 1.0
Average collie weight
For 10 collie weights, sum the weights and divide by 10. The result is about 58.57 pounds. A computer may display a nearby value such as 58.569999..., because floating-point calculations are approximations; rounding gives the practically useful result. section 2.13
collie_weights = [51.2,53.1,53.1,53.9,59.0,
59.3,60.2,61.1,67.2,67.6]
collie_weight_mean = sum(collie_weights) / len(collie_weights)
# Collie Weight Mean (lbs): 58.57000000000001
print("Collie Weight Mean (lbs):", collie_weight_mean)
This short calculation makes the recipe visible: sum the list of weights, divide by its length, and print the resulting mean. section 2.13
The same calculation applies to a sample and a population, but the symbols signal which one is being described. x̄ (“x-bar”) is a sample mean, while μ (“mu”) is a population mean. Individual observed values can be written x₁ through xₙ; Σ means to add them together. section 2.13
Means become especially useful when they connect a dataset to broader statistical questions:
- In a normal distribution, the mean sets the bell curve’s horizontal location; the standard deviation controls its spread. section 4.1 section 4.2
- Across many sizable random samples, the distribution of sample means can approximate a normal distribution even when the population is not normal. section 4.9
- A sample mean can serve as a point estimate for a population mean and anchor a confidence interval with a margin of error. section 5.4
- One-sample Z-tests and t-tests compare a sample mean with a population mean. section 6.3
A mean is a starting point, not the whole story
A survey of 100 people can produce a mean screen time of 7.5 hours per day, but a confidence interval expresses uncertainty more clearly: a 95% interval of 7.01 to 7.99 hours. Treat measured data as incomplete context rather than automatic truth. section 5.4 section 2.21
