Population Mean

Overview

The population mean is the average value of a characteristic in an entire population. It is a key measure of central tendency in statistics and is denoted by $\mu$. When dealing with a sample, the mean of the sample ($\bar{x}$) is used to estimate the population mean.

Formula for Population Mean

$$\mu = \frac{\sum x}{N}$$

Where:

• $\mu$: Population mean.
• $x$: Each data point.
• $N$: Total number of data points in the population.

Sample Mean as an Estimate

When it is impractical to measure an entire population, a sample mean ($\bar{x}$) is used:

$$\bar{x} = \frac{\sum x}{n}$$

Where:

• $\bar{x}$: Sample mean.
• $x$: Each data point in the sample.
• $n$: Number of data points in the sample.

Confidence Intervals for Population Mean

Confidence intervals provide a range in which the population mean likely falls, calculated as:

$$\bar{x} \pm z \times \frac{s}{\sqrt{n}}$$

Where:

• $z$: Z-score corresponding to the desired confidence level.
• $s$: Sample standard deviation.
• $n$: Sample size.

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Worked Examples

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Summary

• Population Mean ($\mu$): Average value across the entire population, calculated as:

$$\mu = \frac{\sum x}{N}$$

• Sample Mean ($\bar{x}$): Used to estimate $\mu$ when the population is too large to measure.
• Confidence Intervals provide a range in which the population mean likely lies, calculated using:

$$\bar{x} \pm z \times \frac{s}{\sqrt{n}}$$

• Population means are fundamental for understanding the central tendency of large data sets.