Take seven measurements: 23, 45, 12, 67, 34, 89 and 21. You can say exactly what they look like. You cannot yet say what the next measurement, or the whole population they came from, will look like. The first job belongs to descriptive statistics. The second belongs to inferential statistics, and mixing them up is how a small sample turns into a confident, wrong conclusion.
Descriptive statistics summarizes the data you have
The goal of descriptive statistics is to summarize a dataset with a small number of measures. Faced with a heap of raw numbers, it asks how we can find the pattern and present it in a digestible way. It does this without assuming anything beyond the dataset itself. The most common summaries are:
- Central tendency: the mean, median, and mode identify the center of the data.
- Dispersion: the variance, standard deviation, and interquartile range (IQR) describe how stretched or squeezed the data are.
- Shape and relationship:
skewnessA measure of asymmetry: how much longer one tail of the distribution is than the other. ,kurtosisA measure of how heavy the tails of a distribution are compared with a normal distribution. , and the linear correlation between two variables.
For our seven measurements, the mean is 41.57 and the standard deviation is 27.71 (dividing by n - 1). Those two numbers describe the sample completely, and they say nothing about anyone outside it.
Inferential statistics estimates what you did not measure
Descriptive statistics describes the exact data on hand. Inferential statistics
generalizes from a sample to a larger population. Measuring a whole
population, such as every person in a country, is often impossible, slow, or
expensive. So researchers measure a sample and use a sampling design and a
statistical model to estimate the population parameters. A large sample alone
does not remove
This works because inferential statistics relies on the laws of probability to reach conclusions and to state how confident we are in them. It uses inductive reasoning: the logic flows from the specific sample up to a broader claim about the population.
The key techniques are:
- Point estimation: a single “best guess” for an unknown population parameter.
- Interval estimation (confidence intervals): an interval built with a specified long-run coverage rate under the model’s assumptions. A 95% procedure covers the fixed population parameter in 95% of repeated samples. It does not assign a 95% probability to that parameter after you observe an interval.
- Hypothesis testing: an assessment of how compatible the data are with a null hypothesis and its assumptions. A p-value is not the probability that the hypothesis is true, and statistical significance does not measure effect size.
From seven numbers to a range for the population mean
Here is the step from description to inference. Assume the seven observations
are an independent random sample from a normal population. A Student’s t
interval uses the sample standard deviation (ddof=1), six mean ± 2.447 * sample_std / sqrt(7). With only seven observations, the
normality assumption matters.
See
The sample mean, 41.57, is a description. The range from 15.9 to 67.2 is an inference. It is wide because seven observations carry little information and the data vary a lot. A larger sample or less variable data would narrow it.
Describe first, then infer, and state your assumptions
Descriptive statistics tells you what is in your sample. Inferential statistics uses probability to say what that sample suggests about the population, and how uncertain that suggestion is. Always describe the data before you infer from it. Then check that your sampling and model assumptions hold, because the interval is only as trustworthy as they are.