6 Descriptive Statistics

What descriptive statistics show

Descriptive statistics summarize observed data so that its typical value, variability, and pattern are easier to understand. They describe the data that were observed; by themselves, they do not establish causes or prove that the same pattern holds in a larger population. Numerical summaries are most useful when interpreted alongside the data’s context and, when appropriate, a graph.

Frequency distributions

A Frequency table organizes counts of how often values or categories appear. For categorical data, it can count labels such as product types or survey responses. When numerical data have many distinct values, they may instead be grouped into intervals: this can make patterns easier to see, but it hides the exact values within each interval.

Relative frequency is the fraction or percentage of all observations in a category:

Cumulative frequency adds counts up to and including a value. Cumulative relative frequency adds the corresponding proportions. For ordered numerical values, these summaries answer questions such as how many observations are at or below a specified value, or what percentage are at or below it. Relative frequencies sum to $1$, or $100\%$, apart from possible rounding.

For the eight observations $2, 3, 3, 4, 4, 5, 6, 13$, the frequency and cumulative summaries are:

• Value $2$: frequency $1$, relative frequency $12.5\%$, cumulative frequency $1$.

• Value $3$: frequency $2$, relative frequency $25\%$, cumulative frequency $3$.

• Value $4$: frequency $2$, relative frequency $25\%$, cumulative frequency $5$.

• Value $5$: frequency $1$, relative frequency $12.5\%$, cumulative frequency $6$.

• Value $6$: frequency $1$, relative frequency $12.5\%$, cumulative frequency $7$.

• Value $13$: frequency $1$, relative frequency $12.5\%$, cumulative frequency $8$.

• Total: frequency $8$, relative frequency $100\%$.

Measures of center

The Mean, Median, and Mode describe a typical or central value, but they do so in different ways.

• Mean: Add all values and divide by their count. For a frequency table, multiply each value by its frequency, add those products, and divide by the total frequency.

• Median: Sort the values. With an odd number of observations, the median is the middle value; with an even number, it is the average of the two middle values.

• Mode: Identify the most frequently occurring value or category. A data set may have one mode, several modes, or no mode if no value occurs more often than the others.

For $2, 3, 3, 4, 4, 5, 6, 13$, the mean is $40 \div 8 = 5$, and the median is $(4 + 4) \div 2 = 4$. Both $3$ and $4$ occur twice, so the data are bimodal. The value $13$ raises the mean, while the median remains $4$; this illustrates why the median can better represent the center when data are skewed or include an extreme value.

Range and quartiles

A measure of center alone does not show whether observations cluster closely or vary widely. Measures of Spread describe variability; common measures include the range, interquartile range, variance, and standard deviation.

The Range is the largest value minus the smallest value. For the example data, $13 - 2 = 11$. It is simple to calculate, but depends only on the two extremes.

Quartiles divide ordered data into four parts. The lower quartile, $Q_1$, is the twenty-fifth percentile; the upper quartile, $Q_3$, is the seventy-fifth percentile. The Interquartile range (IQR) is calculated as:

It describes the spread of the middle half of the data and is less affected by extreme values than the range or standard deviation. Using the median-of-halves convention for the example, $Q_1 = 3$ and $Q_3 = 5.5$, so $\mathrm{IQR} = 5.5 - 3 = 2.5$. Quartile values can differ slightly among software packages because more than one calculation convention is used; state the convention when exact agreement matters.

Variance and standard deviation

Variance summarizes squared distances from the mean, so larger deviations contribute more. Standard deviation is the square root of variance and expresses spread in the same units as the original measurements.

For observations $x_1, x_2, \ldots, x_n$ with mean $\bar{x}$, the population and sample variance formulas are:

Standard deviation is the square root of the corresponding variance. Use the population formula when the data include the entire population of interest. Use the sample formula when the observations are a sample and the goal is to estimate variability beyond that sample.

For the example data, the squared deviations from the mean $5$ sum to $84$. The population variance is $84 \div 8 = 10.5$, with standard deviation about $3.24$. The sample variance is $84 \div 7 = 12$, with standard deviation about $3.46$. Variance is in squared units, whereas standard deviation is in the original units. Like the mean, standard deviation can be strongly affected by extreme observations.

Choosing and interpreting summaries

Choose statistics that fit the data type and shape, and interpret center and spread together.

• For roughly symmetric numerical data without strong outliers, the mean and standard deviation are often informative together.

• For skewed data or data with extreme values, the median and IQR often provide a more resistant description of center and spread.

• For categorical data, report counts and relative frequencies; a numerical mean may have no meaningful interpretation.

Include units and context. A standard deviation of $3.46$ is difficult to interpret unless the reader knows what was measured and on what scale.

A useful workflow is to check the data and measurement units, count observations and inspect missing or unusual values, build frequency summaries, calculate suitable measures of center and spread, and compare the numerical results with a graph. Histograms can reveal distribution shape, center, and spread; box plots emphasize the median, quartiles, and potential extremes. Check that the numerical summaries agree with the observed distribution.