5 Descriptive Statistics

What descriptive statistics reveal

Descriptive statistics organize and summarize observed data. They help answer questions about what value is typical, how much observations vary, and whether values are concentrated, skewed, or divided into groups. A useful summary considers both numerical measures and a graph, because a single statistic can hide important features of a distribution.

Identify the variable type

Choose statistics according to the type of variable. Categorical data place observations into groups, such as transportation method or eye color. Categories are nominal when they have no natural order, or ordinal when they do, as with small, medium, and large. Numerical data represent counts or measurements, such as number of visits or height, and have meaningful magnitudes and differences.

A statistic is useful only when it makes sense for the variable. For example, calculating the average of eye-color labels has no meaningful interpretation.

Summarize categorical data

Summarize categorical data by listing each category with its frequency, or count, and its relative frequency, or share of all observations. In a group of $30$ people, the transportation counts are bus $12$, bicycle $8$, and walk $10$. Their relative frequencies are bus $\frac{12}{30}=40\%$, bicycle $\frac{8}{30}\approx27\%$, and walk $\frac{10}{30}\approx33\%$; the counts total $30$ and the relative frequencies total $100\%$.

The mode is the most frequent category; in this example, it is bus. A frequency or relative-frequency table and a bar chart are effective summaries. For ordinal categories, preserve their natural order and, when useful, report cumulative frequencies. Do not use a histogram for nominal categories: its numerical bins imply intervals that the categories do not have.

Compare measures of center

Measures of center describe a typical value, but they do not always identify the same value or respond equally to unusual observations.

• Mean: Add all values and divide by the number of values. For observations $x_1,\ldots,x_n$, the mean is

  $$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i.$$

  The mean uses every observation, so unusually high or low values can pull it toward a tail.

• Median: Sort the values and find the middle. With an even number of observations, take the mean of the two middle values. The median is less affected by extreme observations than the mean.

• Mode: Use the most frequent value. A data set can have one mode, several modes, or no useful mode; the mode is especially useful for categorical data.

For the ordered data $2,4,4,6,9$, the mean is $5$, the median is $4$, and the mode is $4$. The mean and median need not coincide, especially when values are skewed or include outliers.

Measure numerical spread

Measures of spread describe variation among numerical values.

• Range: The maximum minus the minimum. It is simple to calculate but depends only on the two extremes.

• Interquartile range (IQR): The difference between the third and first quartiles, $\mathrm{IQR}=Q_3-Q_1$. Here, $Q_1$ is the 25th percentile and $Q_3$ is the 75th percentile. The IQR describes the spread of the middle half of the data and is relatively resistant to extreme values.

• Variance: The average squared distance from the mean. For a sample, the usual sample variance is

  $$s^2=\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}.$$

  Squaring gives more influence to values far from the mean, and variance is in squared units.

• Standard deviation: The square root of the variance. It describes spread around the mean in the data’s original units; the sample standard deviation is $s=\sqrt{s^2}$.

Population variance, commonly written $\sigma^2$, instead divides the sum of squared deviations by the population size $N$. Use the sample or population formula appropriate to the data and task.

Read distributions and graphs

A distribution describes the values a variable takes and how often they occur. Describe a numerical distribution by considering its center and spread, its shape, and any unusual features. Shape includes whether the distribution is approximately symmetric or skewed and whether it has one peak or multiple peaks. Unusual features may include gaps, clusters, or possible outliers.

A right-skewed distribution has a longer tail toward larger values; a left-skewed distribution has a longer tail toward smaller values. A histogram helps reveal overall shape, while a box plot compactly displays the median, quartiles, and spread. Graphs and statistics complement one another: a histogram can reveal two clusters that a single mean would conceal.

Use percentiles and identify potential outliers

A percentile indicates a position in ordered numerical data: approximately $p\%$ of observations are at or below the $p$th percentile. The 50th percentile is the median; the 25th and 75th percentiles are the first and third quartiles. A 90th-percentile score is not necessarily 90% correct; it means the score is at or above roughly 90% of scores in the comparison group.

Definitions for calculating percentiles and quartiles can vary slightly among software and textbooks, so use the specified convention when exact values matter. A compact numerical summary is the five-number summary: minimum, $Q_1$, median, $Q_3$, and maximum.

A common screening rule flags observations below $Q_1-1.5\,\mathrm{IQR}$ or above $Q_3+1.5\,\mathrm{IQR}$ as potential outliers. These flags prompt investigation; they do not prove that a value is an error. Check context and data quality before deciding how to handle a flagged value.

Choose a suitable summary

Choose summaries according to variable type. Describe categorical data with category counts, proportions, and often the mode. For numerical data, use measures of center and spread together, and inspect the distribution’s shape and unusual features.

The mean and standard deviation are often useful together for roughly symmetric distributions without strong outliers. For skewed data or data with extreme values, the median and IQR usually give a more representative description. Pairing mean with standard deviation, or median with IQR, makes the choice of summary clear.