Descriptive vs Inferential Statistics

Descriptive statistics summarize the data you actually collected: what is typical, how spread out the values are and what shape the distribution has. Inferential statistics use that sample to draw conclusions about a larger population you did not fully observe, and they attach a measure of uncertainty to those conclusions.

A simple way to tell them apart: if the statement would still be true when you only care about the people in your dataset, it is descriptive. If it makes a claim about people beyond your dataset, it is inferential.

• Descriptive: "The 120 surveyed nurses worked a mean of 41.3 hours per week (SD = 6.2)."
• Inferential: "We estimate that nurses in the region work a mean of 41.3 hours per week, 95% CI [40.2, 42.4]."

The two are not rivals. Every quantitative study reports descriptive statistics first, and inferential results are only interpretable when readers can see the descriptive picture behind them.

Mean, median and mode are the three measures of central tendency: different ways of describing a typical value.

• Mean: the arithmetic average. Add all values and divide by how many there are.
• Median: the middle value when the data are sorted. With an even number of values, it is the average of the two middle values.
• Mode: the most frequent value. A dataset can have one mode, several modes, or none if every value occurs once.

Worked example: how to find the mean, median and mode

Eight students report how many hours they studied last week: 4, 7, 7, 8, 10, 12, 13, 19.

1. Mean: sum = 4 + 7 + 7 + 8 + 10 + 12 + 13 + 19 = 80; 80 / 8 = 10 hours.
2. Median: the values are already sorted. With 8 values, the middle two are the 4th and 5th (8 and 10), so the median is (8 + 10) / 2 = 9 hours.
3. Mode: 7 appears twice and every other value once, so the mode is 7 hours.

Notice the mean (10) is higher than the median (9). The single high value of 19 pulls the mean upward, a sign of right skew. The median is resistant to such extreme values; the mean is not.

Which measure should you use?

Mean, median and mode in biostatistics follow the same logic. Clinical variables such as length of stay, viral load or costs are often heavily right-skewed, which is why medical papers frequently report medians with interquartile ranges rather than means with standard deviations.

For a list of numbers, the mean is:

x̄ = Σx / n

where Σx is the sum of all values and n is the number of values. The population mean uses the same calculation and is written μ = Σx / N.

Mean from a frequency table

If the data come as values with frequencies, multiply each value by its frequency, add the products and divide by the total frequency: x̄ = Σ(f × x) / Σf. For example, if 3 students read 1 book, 5 read 2 books and 2 read 4 books, the mean is (3×1 + 5×2 + 2×4) / 10 = 21 / 10 = 2.1 books.

How to find the mean of a probability distribution

For a discrete random variable, the mean (also called the expected value) weights each value by its probability: μ = Σ[x × P(x)]. Suppose X is the number of library books a student borrows in a week, with P(0) = 0.2, P(1) = 0.5 and P(2) = 0.3. Then:

μ = 0(0.2) + 1(0.5) + 2(0.3) = 0 + 0.5 + 0.6 = 1.1 books

The variance is σ² = Σ[x² × P(x)] − μ² = (0 + 0.5 + 1.2) − 1.21 = 0.49, so the standard deviation is σ = 0.7 books.

Standard deviation is a measure of how spread out values are around the mean. In simple terms, it is roughly the typical distance of a data point from the average. A small standard deviation means values cluster tightly around the mean; a large one means they are widely scattered.

Two datasets can have the same mean but very different standard deviations. Exam scores of 68, 70 and 72 and scores of 40, 70 and 100 both average 70, but the first set is consistent and the second is not. Reporting the mean without the standard deviation hides that difference, which is why "mean and standard deviation" almost always appear together.

Variance and standard deviation formulas

Variance is the average squared deviation from the mean; standard deviation is its square root, which brings the measure back to the original units.

Sample standard deviation vs population standard deviation

Use the population formula only when your data contain every member of the group you care about, such as all 30 students in one class when you are describing just that class. Use the sample standard deviation (s) when your data are a sample used to learn about a larger population, which is almost always the case in research.

The sample formula divides by n − 1 instead of n (Bessel's correction). Because deviations are measured from the sample mean, which sits in the middle of your particular data, they tend to be slightly smaller than deviations from the true population mean. Dividing by n − 1 corrects this, making s² an unbiased estimate of the population variance. Statistical software reports the sample version by default.

Using the same study-hours data (4, 7, 7, 8, 10, 12, 13, 19) with mean 10, here are the steps of standard deviation:

1. Find the mean: x̄ = 10.
2. Subtract the mean from each value (second column). These deviations always sum to zero, which is a useful check.
3. Square each deviation (third column) and add them: Σ(x − x̄)² = 152. This is the sum of squares.
4. Divide by n − 1 to get the sample variance: s² = 152 / 7 = 21.71.
5. Take the square root: s = √21.71 = 4.66 hours.

If these eight students were the entire population of interest, you would divide by N = 8 instead: σ² = 152 / 8 = 19.0 and σ = √19 = 4.36 hours. The difference between s and σ shrinks as the sample gets larger.

How to interpret the standard deviation

Here, students studied an average of 10 hours, typically varying by about 4.7 hours from that average. For data that are roughly bell-shaped, a standard deviation chart follows the empirical rule: about 68% of values fall within one standard deviation of the mean and about 95% within two. With skewed data like ours (one student at 19 hours), that rule is only approximate, and the median with the interquartile range may describe the data better. The guide to the normal distribution explains the 68-95-99.7 rule in detail.

A useful habit when analyzing standard deviation is to compare it with the mean and the scale. An SD of 4.7 hours is large relative to a mean of 10 (substantial variation between students), whereas an SD of 4.7 points on a 0 to 100 exam with mean 75 would indicate fairly consistent performance.

Central tendency and standard deviation are the core, but quantitative descriptive analysis usually includes several other measures:

• Range: maximum minus minimum (19 − 4 = 15 hours in our example). Simple, but determined entirely by the two most extreme values.
• Interquartile range (IQR): the spread of the middle 50% of values (Q3 − Q1). Pair it with the median for skewed data. Note that software packages use slightly different quartile methods, so small differences between programs are normal.
• Frequencies and percentages: the main description for categorical variables, such as "62% of participants were first-year students."
• Shape: skewness and kurtosis describe asymmetry and tail weight.

Charts for descriptive data

• Histograms and boxplots for the distribution of a numeric variable.
• Bar charts for frequencies of categories.
• Means with error bars to compare groups, always stating whether the bars show the standard deviation, standard error or a confidence interval.

Descriptive statistics example table

A table like this, usually Table 1 of a results chapter, gives readers the descriptive picture before any tests. Our guide to data analysis in research covers how this fits into a full analysis plan.

Inferential statistics answer the question: given this sample, what can I reasonably say about the population? Because a different random sample would give slightly different numbers (sampling error), every inferential conclusion comes with an estimate of uncertainty. The two main tools are estimation, mainly confidence intervals, and hypothesis testing, which produces test statistics and p-values.

Inference is only as good as the sample. Random sampling, or random assignment in experiments, is what justifies generalizing. A convenience sample of volunteers from one university may describe those volunteers perfectly well, but inferential claims about "all students" then rest on assumptions you should state openly.

Confidence intervals: a worked example

A confidence interval gives a range of plausible values for a population parameter. For a mean:

CI = x̄ ± t* × (s / √n)

where t* is the critical value from the t distribution with n − 1 degrees of freedom.

1. From the study-hours sample: x̄ = 10, s = 4.66, n = 8.
2. Standard error: SE = 4.66 / √8 = 1.648.
3. Critical value for 95% confidence with df = 7: t* = 2.365.
4. Margin of error: 2.365 × 1.648 = 3.90.
5. 95% CI: 10 ± 3.90, or 6.10 to 13.90 hours.

Interpretation: we can be 95% confident that the mean weekly study time in the population these students were sampled from lies between about 6.1 and 13.9 hours. Strictly, "95% confident" means the method captures the true mean in 95% of repeated samples. The interval is wide because n = 8 is small; quadrupling the sample size would roughly halve it.

Hypothesis testing: a quick example

Suppose a university guideline says students should study 8 hours a week. Is our sample's mean of 10 evidence that students in general study more? A one-sample t-test gives t = (10 − 8) / 1.648 = 1.21 with df = 7 and p = .26. The difference is not statistically significant: a sample mean this far from 8 would occur fairly often by chance with only 8 students. Note that the 95% CI (6.10 to 13.90) contains 8, which tells the same story. The t-test guide walks through this calculation in detail.

The right test depends on your research question, the measurement level of your variables and your design. Our guide to choosing a statistical test walks through that decision step by step.

Each example below starts with descriptive statistics for the sample and then makes an inferential claim about a population. All are illustrative scenarios.

1. Education: A researcher randomly assigns 60 students to a flipped classroom or a lecture format. Descriptive: mean exam scores are 78 and 72. Inferential: an independent-samples t-test assesses whether the 6-point difference is likely to hold for students in general, and a confidence interval estimates its size.
2. Public health: A survey of a random sample of 500 adults finds 31% report regular exercise. Inferential: a 95% confidence interval for the population proportion is about 31% ± 4 percentage points (using p ± 1.96√[p(1 − p)/n]).
3. Psychology: Participants rate their stress before and after a mindfulness program. Inferential: a paired t-test (or Wilcoxon signed-rank test if the differences are skewed) evaluates whether stress decreases in the wider population the participants represent.
4. Business: An analyst examines whether customer satisfaction differs across four store locations using one-way ANOVA, followed by post hoc comparisons.
5. Biostatistics: A clinical study uses logistic regression to estimate how the odds of readmission change with age and comorbidities, reporting odds ratios with 95% confidence intervals.
6. Sociology: A chi-square test examines whether employment status is associated with housing type in a national survey sample.

Check example 2: with p = .31 and n = 500, the standard error is √(.31 × .69 / 500) = .0207, and 1.96 × .0207 = .041, so the margin is about 4.1 percentage points.

• Describe before you infer. Report sample size, means and standard deviations (or medians and IQRs) for every variable used in tests.
• Match the descriptive measure to the data. Mean and SD for roughly symmetric data; median and IQR for skewed data; counts and percentages for categories.
• Report effect sizes and intervals, not just p-values. "p < .05" alone says nothing about how large or important a difference is.
• Keep the language honest. Do not describe a non-significant result as "no difference," and do not generalize beyond the population your sample represents.
• Use consistent decimals. Two decimals for most statistics is a common convention in APA style; report p-values to two or three decimals.

A typical sentence combining both: "Students in the flipped classroom scored higher (M = 78.0, SD = 8.1) than those in the lecture format (M = 72.0, SD = 9.0)," followed by the test statistic, p-value, effect size and confidence interval for the difference.