This page is a working formula sheet for introductory statistics. The table above lists each formula next to the same formula filled in with the numbers in the data box, so you can see exactly how every symbol turns into arithmetic. Change the data, or narrow the sheet to one family of formulas, and everything updates.
How to use this formula sheet
- Leave the sample data in place or paste your own numbers (at least two).
- Use Show formulas for to focus on center, spread, position or shape, or keep Everything.
- Read across each row: the measure, its formula, the values substituted from your data, and the result.
Notation
| Symbol | Meaning |
|---|---|
| n, N | number of values in a sample, a population |
| Σ | add up over every value |
| x̄, μ | sample mean, population mean |
| s, σ | sample and population standard deviation |
| Q1, Q3 | first and third quartiles |
| z | standard score |
Measures of center
With the sample data 4, 7, 7, 9, 12, 15: the sum is 54, so x̄ = 54 ÷ 6 = 9; the median is the average of 7 and 9, 8; and the mode is 7.
Measures of spread
For the sample data the deviations from 9 are −5, −2, −2, 0, 3, 6; their squares add to SS = 78. So s² = 78 ÷ 5 = 15.6 and s ≈ 3.9497, while the population versions are σ² = 13 and σ ≈ 3.6056. Q1 = 7 and Q3 = 12, so the IQR is 5.
Measures of position
The largest sample value, 15, has z = (15 − 9) ÷ 3.9497 ≈ 1.52. The fences are 7 − 7.5 = −0.5 and 12 + 7.5 = 19.5, so no value is an outlier.
Measures of shape
Positive skewness means a longer right tail. A quick alternative is Pearson’s median skewness, 3(x̄ − median) ÷ s, which is about 0.76 for the sample data.
Sampling and inference
These formulas are not evaluated in the table, but they build directly on it:
For a 95% interval, z* = 1.96; t* comes from a t-table with n − 1 degrees of freedom (2.571 for the six sample values). For the sample data, SE ≈ 3.9497 ÷ √6 ≈ 1.6125, so a 95% interval for the mean is about 9 ± 2.571 × 1.6125, or 4.85 to 13.15.
Different textbooks and software use different quartile and percentile conventions. This sheet uses the median-of-halves quartiles found on TI-84 calculators; the quartile calculator shows how Excel's methods compare.
For full step-by-step work on a single measure, open the standard deviation calculator, the variance calculator or the z-score calculator; for normal probabilities use the normal distribution calculator.
Frequently asked questions
What is the difference between sample and population formulas?
Population formulas use Greek letters (μ, σ) and divide sums of squares by N. Sample formulas use Latin letters (x̄, s) and divide by n − 1 so that the sample variance is an unbiased estimate of the population variance.
Which formulas do I need for an intro statistics exam?
Usually the mean, median, sample variance and standard deviation, z-scores, the five-number summary with the 1.5 × IQR rule, the standard error s ÷ √n and a confidence interval for a mean or proportion. All of them are on this sheet.
Why does the calculator need at least two values?
The sample variance divides by n − 1, which is zero for a single value, and every formula that uses s depends on it. With two or more numbers every formula on the sheet can be evaluated.
Which quartile method is used on this sheet?
The median-of-halves method, which excludes the median from both halves when n is odd. It matches TI-83 and TI-84 calculators and most US textbooks; the quartile calculator compares it with Excel's methods.