6.2: Descriptive Statistics - Summarizing Data

Descriptive Statistics - Summarizing Data

Descriptive statistics are methods for summarizing and describing the essential features of a dataset. When faced with a large collection of numbers, descriptive statistics help reduce the data to a few understandable figures that capture key properties of the distribution. The most commonly used descriptive measures include those of central tendency (e.g. mean, median, mode) and measures of variability or spread (e.g. range, standard deviation, variance, percentiles). These measures allow analysts to quickly communicate what is “typical” in the data and how much the data points differ from each other.

Measures of Central Tendency

Central tendency refers to the middle or typical value of a dataset. The three primary measures are:

• Mean – the arithmetic average of all values. The mean is calculated by summing all observations and dividing by the number of observations. For a dataset with values $x_1, x_2, \dots, x_n$, the mean is:

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

It is a very common summary, but can be sensitive to extreme values (outliers). For example, in {10, 15, 22, 60}, the mean is $(10+15+22+60)/4 = 26.75$. A key point is that the mean uses all data points, so it can be skewed by a single very large or small observation.

• Median – the middle value that divides the sorted data into two equal halves. If $n$ is odd, the median is the middle observation; if $n$ is even, it is the average of the two middle observations. The median is resistant to outliers, meaning it’s not affected much by extreme values. Using the previous example {10, 15, 22, 60} (sorted), the median is the average of 15 and 22, which is 18.5. If a dataset is skewed (asymmetric), the median is often a better representation of a "typical" value than the mean.
• Mode – the most frequently occurring value in the dataset. In a set of discrete or categorical data, the mode identifies the common category. Some datasets can be bimodal or multimodal (having two or more modes) if multiple values tie for highest frequency. For example, in {red, blue, blue, green}, “blue” is the mode since it appears most often. Mode is less used for quantitative data unless the distribution has clear peaks.

Each of these measures gives a different perspective: the mean captures the balance point of the data (useful in many calculations), the median gives the middle and is more robust to skewed data, and the mode indicates the most common value. In a perfectly symmetric distribution (like a perfect normal distribution), the mean = median = mode, all at the center. In skewed distributions, these can differ – for instance, a right-skewed (long tail to the right) distribution typically has mean > median > mode.

Example (Python): Let’s illustrate calculating some of these measures using Python:

import numpy as np
data = [10, 15, 22, 60]  # example dataset
mean_val = np.mean(data)
median_val = np.median(data)
# Standard deviation (population version by default, ddof=0)
std_val = np.std(data) 
print("Mean:", mean_val)
print("Median:", median_val)
print("Std Dev:", std_val)

Figure \(\PageIndex{1}\): Measures of Central Tendency in Skewed vs. Symmetric Distributions. Three distribution curves illustrate how mean, median, and mode align in a symmetric distribution but diverge in left-skewed and right-skewed distributions. (CC BY 4.0; Felix Amoruwa). Access a detailed description of Figure \(\PageIndex{1}\)

Running this code would output the mean as 26.75, the median as 18.5, and the standard deviation (a measure of spread, discussed below) as around 19.98. For a quick summary of a dataset, we can also use pandas:

import pandas as pd
df = pd.DataFrame(data, columns=['Values'])
print(df['Values'].describe())

This would display count, mean, std, min, 25th percentile (Q1), 50th percentile (median), 75th percentile (Q3), and max. Such summaries make it easy to see central values and the range of the data at a glance.

Measures of Dispersion (Variability)

Besides the center, we need to describe how spread out the data are. Two datasets might have the same mean but very different variability. Key measures of dispersion include:

• Range – the difference between the maximum and minimum values. It gives a crude sense of spread (Range = max – min). In our example {10, 15, 22, 60}, the range is 60–10 = 50. However, the range uses only two data points and can be overly influenced by extremes.
• Variance – the average of the squared deviations from the mean. For a population of size $N$, the variance is $\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2$ (where $\mu$ is the population mean). For a sample of size $n$, the sample variance is $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$ (using $n-1$ in the denominator to correct bias in estimating the population variance). Variance is in squared units of the original data, which can be less intuitive. In the example, the sample variance would be $\frac{1}{3}[(10-26.75)^2 + \dots + (60-26.75)^2] \approx 399.17$.
• Standard Deviation (SD) – the square root of variance. This brings the dispersion back to the original units. SD is perhaps the most commonly reported measure of spread, as it is easier to interpret: it roughly indicates the “typical” distance of data points from the mean. A small SD means data are tightly clustered around the mean; a large SD means data are more spread out. In our example, the (sample) standard deviation $s \approx \sqrt{399.17} \approx 19.98$. If a distribution is approximately normal (bell-shaped), about 2/3 of values lie within 1 SD of the mean, and about 95% within 2 SDs (this is the empirical rule, discussed later) [15].
• Interquartile Range (IQR) – the range of the middle 50% of data, defined as Q3 – Q1 (the 75th percentile minus the 25th percentile). It measures the spread of the central portion of the data and is more robust to outliers. A box plot is a common visualization that shows the median and the IQR (the “box”), with whiskers often extending to 1.5×IQR to indicate a typical range, beyond which points are considered outliers.
• Percentiles – indicate the value below which a given percentage of data falls. For example, the 25th percentile (first quartile, Q1) is the value such that 25% of observations are less than or equal to it; the 90th percentile is the value below which 90% of data lie. Percentiles are useful for understanding the distribution of data (e.g. median is 50th percentile). In standardized exams, scoring in the 90th percentile means you scored higher than 90% of test-takers. Many summary statistics reports will list percentiles to show the spread (e.g. 5th, 25th, 75th, 95th).

In summary, measures of dispersion complement measures of center by describing the variability in data. For instance, two stores might have the same average daily sales (say \$500), but if one store’s daily sales range from \$100 to \$900 (high variance) and the other ranges only \$400 to \$600 (low variance), the business implications differ greatly. Analysts often look at both center and spread: Are the data tightly clustered or widely scattered? Are there outliers? The combination of mean and standard deviation (or median and IQR for skewed data) provides a concise description of the data’s distribution.

Figure \(\PageIndex{2}\): Comparing Datasets with Same Mean but Different Variability. Two dot plots or histograms show datasets sharing the same mean but with low versus high standard deviation, illustrating why measures of dispersion matter. (CC BY 4.0; Felix Amoruwa) Access a detailed description of Figure \(\PageIndex{2}\)

(In our code example above, np.std returned about 19.98, reflecting the spread in the numbers. The pandas .describe() output would also list the 25% and 75% quartiles, making it easy to compute IQR.)

Data Distribution Shape

Beyond single numbers, it’s often useful to consider the shape of the data distribution. This includes assessing whether the distribution is symmetric or skewed, and whether it is unimodal (one peak) or multimodal. Two important shape characteristics:

• Skewness – A distribution is skewed right (positively skewed) if it has a long tail on the right (high values are the tail), often causing mean > median. It is skewed left (negatively skewed) if the long tail is on the left (mean < median). For example, income distributions in many countries are right-skewed: a small number of people have extremely high incomes, pulling the mean above the median. In skewed data, median is usually a better measure of center for reporting, and the difference between mean and median can indicate the direction of skew.
• Modality – the number of peaks in the distribution. A single peak is unimodal. Two peaks might indicate a mixture of two populations (e.g. test scores might be bimodal if one group of students was well-prepared and another not at all). The mode(s) can be seen as peaks in a histogram or density plot.

Visual tools like histograms and box plots are invaluable for assessing distribution shape. A histogram groups data into bins and shows how many fall into each bin, giving a sense of the frequency distribution. A density plot or curve can smooth out a histogram to show an approximate continuous distribution shape. We will see some specific distribution shapes (like the bell curve of the normal distribution) in the next section.

Understanding the shape helps in choosing the right summary statistics and analytic approach. For instance, if data are highly skewed, one might consider a logarithmic transformation to normalize it, or use non-parametric statistical methods that don’t assume normality. Always remember to look at the data (through plots) in addition to summarizing it with statistics – the combination provides the fullest picture.