Introduction

In the world of statistics, one of the fundamental tasks is to estimate key characteristics of a population using sample data. This chapter focuses on estimating three crucial measures: the mean, variance, and standard deviation. These measures provide insights into the central tendency and dispersion of a dataset. By mastering the art of estimation, you’ll be equipped to draw meaningful conclusions from sample data, bridging the gap between limited observations and broader population characteristics.

Estimating the Mean

The mean, often referred to as the average, is a measure of central tendency. To estimate the mean using sample data, sum up all the values and divide by the number of observations. The formula for the sample mean is:

• Sample Mean $=\frac{\sum \text{ Values }}{\text{ Number of Observations }}$

Example: Suppose we have a sample of test scores: $85,90,78,92$, and 88 . The sample mean is calculated as:

• Sample Mean $=\frac{85+90+78+92+88}{5}=86.6$

Estimating the Variance and Standard Deviation

The variance measures the spread or dispersion of data points around the mean. The standard deviation is the square root of the variance and provides a more interpretable measure of spread. To estimate the variance and standard deviation, use the following formulas:

• Sample Variance $=\frac{\sum (x-\overline{x}{)}^2}{n-1}$
• Sample Standard Deviation $=\sqrt{\text{ Sample Variance }}$

Here, $x$ represents each data point, $\overline{x}$ is the sample mean, and $n$ is the number of observations.

Example: Using the same test scores, the sample mean (calculated earlier) is 86.6. The sample variance and standard deviation can be calculated as:

• Sample Variance $=\frac{(85-86.6{)}^2+(90-86.6{)}^2+(78-86.6{)}^2+(92-86.6{)}^2+(88-86.6{)}^2}{5-1}=31.7$
• Sample Standard Deviation $=\sqrt{31.7}\approx 5.63$