7 Descriptive Statistics

What descriptive statistics show

Descriptive statistics organize and summarize observed data to answer questions such as which category occurs most often or how much measurements vary. They describe the data at hand; by themselves, they do not establish causes or prove that a pattern will hold for a larger population.

Summarizing categorical data

Categorical data assign observations to groups, such as product type, region, or satisfaction rating. For each category, calculate its frequency, the count of observations, and its relative frequency, the proportion of all observations:

For example, if 12 of 30 survey responses are “Online,” the relative frequency is $\frac{12}{30} = 0.40$, or $40\%$. A frequency table makes category comparisons easy, and a bar chart can display counts or percentages.

Use consistent category labels. If labels are not standardized, “Online” and “online” may be counted separately. When calculating a percentage, be clear about the denominator, especially if some responses are blank or missing.

In a spreadsheet, if responses are in A2:A31 and a category name is in C2, use:

Divide the count by the number of valid responses to get the percentage. COUNTIF counts cells that match one criterion; use COUNTIFS when counting observations that meet multiple criteria.

Measures of center

Numerical data represent quantities, such as time, age, or sales. A useful summary describes both center, or a typical value, and spread, or how much values differ.

The mean is the sum of all values divided by the number of values. It uses every observation, so unusually high or low values can pull it toward them. The median is the middle value after sorting; with an even number of observations, it is the mean of the two middle values. It is generally less affected by extreme values than the mean.

The mode is the most frequently occurring value. A data set can have more than one mode, or none if no value repeats. For categorical data, the mode is the category with the highest frequency.

For the values $12, 14, 14, 15, 20$, the mean is $15$, the median is $14$, and the mode is $14$. The high value of $20$ raises the mean above the median.

Measures of spread

The range is the maximum minus the minimum. It is simple to calculate, but depends only on the two extremes. For the example values $12, 14, 14, 15, 20$, the range is $20 - 12 = 8$.

The first and third quartiles, $Q_1$ and $Q_3$, mark the 25th and 75th percentiles. The interquartile range (IQR) is $Q_3 - Q_1$ and describes the spread of the middle half of the observations.

Standard deviation measures the typical distance of values from their mean. A small standard deviation means values are relatively clustered; a large one means they are more dispersed. It is sensitive to extreme values.

Spreadsheet functions and conventions

Choose the standard deviation version according to what the data represent. Use the sample standard deviation when observations are a sample used to estimate a larger population. Use the population standard deviation when the data include the entire population of interest.

Assuming example values are in cells B2:B6, common spreadsheet functions include:

STDEV.S calculates sample standard deviation; STDEV.P calculates population standard deviation. The last two formulas calculate $Q_1$ and $Q_3$; subtract the first result from the second to obtain the IQR. Spreadsheet programs may offer different quartile conventions, so state the method when exact comparisons matter. Function names and argument separators can also differ by application or regional settings.

A practical analysis workflow

1. Identify the unit and variables. Determine what each row represents and whether each variable is categorical or numerical.

2. Check the data. Look for missing entries, duplicate records, inconsistent category labels, numbers stored as text, and values outside plausible ranges. Do not silently treat missing values as zero.

3. Choose suitable summaries. For categories, report counts and percentages. For numerical values, report a measure of center and a measure of spread. Use the median and IQR when extreme values or skew make the mean and standard deviation misleading; consider reporting both when useful.

4. Inspect a display. Use a bar chart for category counts or percentages and a histogram for the distribution of numerical values. A histogram can reveal shape, skew, clusters, and potential outliers.

5. Interpret in context. Include units, the number of observations, the denominator for percentages, and any important exclusions or data limitations. A statistic is a summary, not an explanation of why a pattern occurred.