Sample Mean
x̄ = Σxᵢ / n
The sample mean is the arithmetic average of a set of observations. It serves as the most common measure of central tendency and is used as a point estimator for the population mean. The sample mean minimizes the sum of squared deviations from any constant.
Variables
The arithmetic average of the observed values
The sum of all individual data values in the sample
The total number of observations in the sample
Example Calculation
Scenario
A professor records exam scores for 6 students: 78, 85, 92, 88, 74, and 91. Calculate the sample mean.
Given Data
Calculation
x̄ = Σxᵢ / n = 508 / 6
Result
x̄ = 84.67
Interpretation
The average exam score for these 6 students is approximately 84.67 points. This value can be used to estimate the population mean score for all students who might take this exam.
When to Use This Formula
- ✓Summarizing the central tendency of a data set
- ✓Estimating the population mean from a sample
- ✓Calculating other statistics that depend on the mean, such as variance and standard deviation
- ✓Comparing groups in hypothesis testing
Common Mistakes
- ✗Confusing the sample mean (x̄) with the population mean (μ)
- ✗Including outliers without considering their impact on the mean
- ✗Using the mean for heavily skewed distributions where the median would be more appropriate
- ✗Dividing by the wrong count when data is grouped or weighted
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Common questions about this formula
The sample mean (x̄) is calculated from a subset of the population and is used as an estimate of the population mean (μ). The population mean is the true average of every individual in the entire population, which is often unknown. As sample size increases, the sample mean converges to the population mean by the law of large numbers.
Use the median when your data is heavily skewed or contains extreme outliers. The mean is sensitive to outliers because every value contributes to the sum. For example, in income data where a few very high earners inflate the total, the median provides a better measure of the typical value.