A distribution is summarized by a measure of center (mean or median) and a measure of spread (standard deviation, IQR, or range). Edit the data, drag one value far out, and watch which statistics move and which ones resist. Every calculation is worked out step by step.
Type numbers separated by commas or spaces, or start from a preset.
Converting units means every value becomes new = a + b × old. Adding a shifts center but not spread. Multiplying by b scales both.
The mean is the balance point: add the values and divide by n. The median is the middle value of the ordered data. With an odd n it is one data value; with an even n it is the average of the middle two.
The median and IQR are resistant: one extreme value barely moves them. The mean, standard deviation, and range are not resistant, because they use the actual size of every value.
In a right-skewed distribution the long tail pulls the mean above the median. In a left-skewed distribution the mean is below the median. When the distribution is roughly symmetric the two are close.
The standard deviation is roughly the typical distance of the values from the mean. It is never negative, and it equals 0 only when every value is the same. Its units match the data's units.
Q₁ is the median of the lower half and Q₃ the median of the upper half. The IQR = Q₃ − Q₁ is the range of the middle 50% of the data. An outlier is any value below Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR.
Report the mean with the standard deviation when the distribution is roughly symmetric with no outliers. Report the median with the IQR when it is skewed or has outliers.
What each summary statistic measures, and how it reacts.
| Statistic | Measures | Resistant? | Add a constant a | Multiply by b (> 0) |
|---|---|---|---|---|
| Mean | Center | No | Increases by a | Multiplied by b |
| Median | Center | Yes | Increases by a | Multiplied by b |
| Q₁, Q₃, min, max | Position | Quartiles yes; min and max no | Increase by a | Multiplied by b |
| Standard deviation | Spread around the mean | No | No change | Multiplied by b |
| IQR | Spread of the middle 50% | Yes | No change | Multiplied by b |
| Range | Total spread, max − min | No | No change | Multiplied by b |
| Variance s² | Spread, in squared units | No | No change | Multiplied by b² |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 1.7. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 1 terminology guide.