1. What Does Statistics Do?
Statistics provides methods for organizing data, describing its main features, and making estimates or decisions. Descriptive statistics summarizes observed data, while inferential statistics uses a sample to learn about a larger population.
| Area | Purpose | Examples |
|---|---|---|
| Descriptive statistics | Summarize observed data | Mean, median, mode, variance, standard deviation |
| Inferential statistics | Estimate or test population characteristics | Confidence intervals, p-values, hypothesis tests, regression |
2. Mean, Median, and Mode
| Measure | Meaning | Best used for |
|---|---|---|
| Mean | Sum divided by the number of observations | Numeric data without influential outliers |
| Median | Middle value after sorting | Skewed data such as income or prices |
| Mode | Most frequent value | The most common value or category |
For the waiting times 2, 3, 3, 4, 18:
Mean = (2 + 3 + 3 + 4 + 18) ÷ 5 = 6
The median is 3 and the mode is 3. The unusually long 18-minute wait pulls the mean upward, while the median and mode remain unchanged.
3. Variance, Standard Deviation, and Sigma
Measures of center do not describe spread. The datasets 4, 5, 6 and 0, 5, 10
both have a mean of 5, but the second is much more dispersed.
| Term | Meaning |
|---|---|
| Deviation | Difference between an observation and the mean |
| Variance | Average squared deviation |
| Standard deviation | Square root of variance, expressed in the original unit |
| σ (sigma) | A symbol commonly used for population standard deviation |
For 2, 4, 6, the mean is 4 and the squared deviations are 4, 0, 4.
Population variance σ² = (4 + 0 + 4) ÷ 3 ≈ 2.67
Population standard deviation σ = √(8/3) ≈ 1.63
Dividing by n describes a complete population or the observed dataset itself.
Estimating population variance from a sample commonly uses n−1.
4. P-values and Hypothesis Tests
A p-value is the probability of obtaining a result at least as extreme as the observed result, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true.
- State the null hypothesis.
- Choose a significance level before inspecting the result.
- Select a test that matches the design and data.
- Calculate the test statistic and p-value.
- Reject the null hypothesis when
p < α.
The common 0.05 threshold is a convention, not a law of nature. Statistical significance is also different from practical importance. Examine effect sizes, confidence intervals, sample size, assumptions, and plots together with the p-value.
5. Regression Analysis
Regression expresses the relationship between an explanatory variable x
and a response variable y. Simple linear regression uses:
Predicted value ŷ = a + bx
The intercept a is the prediction at x=0, and the slope b
is the average change in y for a one-unit increase in x.
A regression association does not by itself establish causation.
6. The Least-Squares Method
A residual is the difference eᵢ = yᵢ − ŷᵢ between an observed and predicted value.
Ordinary least squares chooses the intercept and slope that minimize:
Sum of squared residuals = Σ(yᵢ − ŷᵢ)²
Squaring prevents positive and negative errors from cancelling, but also makes the fit sensitive to outliers. Inspect the slope, R², p-value, residuals, and uncertainty rather than judging a model from one number.
7. A Practical Order for Reading Data
- Check sample size, missing values, and input errors.
- Summarize the center with the mean, median, or mode.
- Measure spread with variance and standard deviation.
- Inspect distributions and outliers with appropriate plots.
- Choose a hypothesis test or regression model that matches the question.
- Report uncertainty, effect sizes, and diagnostics—not only p-values.
8. Check the Mean, Median, and Standard Deviation in Sakura Chart
Use this waiting-time dataset to reproduce the measures in this article and see how an outlier changes the result.
The LEVEL column separates weekdays from weekends.
- On the preview page, choose Σ (Statistics) as the chart type.
- Select
wait_minutes, then choose Count, Missing, Mean, Median, Q1, Q3, Min, Max, and Sigma. - To compare weekdays and weekends, set the LEVEL column to
LEVELand enable the group summary. - Run the calculation to see overall and grouped tables. The result can also be saved to Excel.
How to read the overall result
Count is 5, Missing is 0, Mean is 6, Median is 3, Q1 is 3, Q3 is 4, Min is 2, and Max is 18. The gap between the mean of 6 and median of 3 shows how the 18-minute wait pulls the mean upward.
Sigma on this screen is the sample standard deviation, calculated with n−1, and is approximately 6.745.
It differs from population standard deviation, which divides by n, so state which definition you used when reporting a result.
Statistics mode summarizes the values. To inspect the distribution and locate the 18-minute observation, create a box plot or histogram from the same data.
9. Key Takeaways
- The mean, median, and mode describe the center in different ways.
- Variance, standard deviation, and sigma describe spread.
- A p-value is defined under the assumption that the null hypothesis is true.
- Regression represents relationships between variables.
- Least squares minimizes the sum of squared residuals.
- Always inspect the data and the model assumptions.
References: NIST/SEMATECH Statistical Glossary, NIST/SEMATECH: Critical Values and p Values