Descriptive Statistics Practice Mean Median Variance and Standard Deviation

Descriptive statistics summarize the center and spread of a set of observations. This guide adapts the topics in the supplied Chapter 2 practice document into a worked reading resource. The calculations below distinguish sample and population measures explicitly rather than reproducing unverified answer-key values.

Start with the research question

Before calculating, identify what each observation represents, its units, the number of observations, and whether the observations are the whole population of interest or a sample. These decisions affect the interpretation of an average and the denominator used for variance.

Mean median and mode

• The arithmetic mean is the sum of the values divided by the number of values. It uses every observation and can be influenced by extreme values.
• The median is the middle value after sorting the observations. With an even number of observations, average the two middle values.
• The mode is the most frequent value. A dataset can have more than one mode, or no uniquely most frequent value.

For the five observations 8, -5, 7, -10 and 5, the sum is 5. Dividing by 5 observations gives a mean of 1. Sorting gives -10, -5, 5, 7, 8, so the median is 5. Every value appears once, so there is no unique mode.

Calculate the sum of squared deviations

Subtract the mean from each value and square the result. For this example, the deviations are 7, -6, 6, -11 and 4. Their squares are 49, 36, 36, 121 and 16. The sum of squared deviations is 258.

Choose sample or population variance

Population variance divides the sum of squared deviations by n. Sample variance divides by n - 1 when estimating population variance from a sample. State which convention is used; the two answers serve different purposes and should not be mixed.

Interpret standard deviation in the original units

Standard deviation is the square root of variance. If observations are measured in a particular unit, standard deviation has that same unit, while variance has squared units. A larger standard deviation indicates a wider spread around the mean; it does not by itself explain the cause of that spread.

Practice explaining results

• Create a small dataset with an extreme value and compare its mean and median. Explain why the two summaries differ.
• Compare two groups using both a measure of center and a measure of spread. Similar averages do not imply similar distributions.
• Check calculations using a spreadsheet, calculator or statistical package, and verify whether its variance function uses the sample or population convention.
• Write the result in plain language, including units, sample size and the limits of the data. Avoid treating a small classroom example as evidence about a wider population.

A final calculation check

Confirm that all observations were copied correctly, negative signs were retained, values were sorted before finding the median, and rounding was postponed until the last step. Report the convention and the precision used so another reader can reproduce the result.