# Data Distributions and Skewness

Psychology I · Statistics · https://tryals.app/learn/psychology-i/data-distributions-and-skewness

## When the Bell Curve Leans

Not all data distributions are symmetric. **Skewness** measures the direction and degree of asymmetry in a distribution.

A **positively (right) skewed** distribution has a long tail stretching toward higher values, like household income. A **negatively (left) skewed** distribution has a long tail toward lower values, like retirement age.

| Skew direction | Tail points | Mean vs. median |
| --- | --- | --- |
| Positive (right) skew | Toward higher values | Mean > median |
| Negative (left) skew | Toward lower values | Mean < median |
| No skew (symmetric) | No tail | Mean = median |

**Common pitfall:** Skew is always named for where the **tail** points, not where the bulk of the data sits. A positively skewed distribution has most of its data on the left.

## Skew and Central Tendency

Skew directly affects summary statistics. The **mean is pulled toward the tail** by extreme values, while the median remains stable.

For skewed data, the median better represents the typical case. **Kurtosis** is a separate property that describes how sharply peaked or flat a distribution is, independent of its skew.

$$\text{Positive skew: mean} > \text{median}$$

$$\text{Negative skew: mean} < \text{median}$$

Always check your distribution's shape before choosing whether to report the mean or the median.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. In a right-skewed income distribution the mean sits above the median.

**Answer:** True

**Why:** True. The mean is computed from every value and is dragged upward by the long right tail, while the median only counts positions and is not. That is why median income is the honest headline figure and mean income is the one that flatters, the gap between them measures the skew.

Page: https://tryals.app/practice/psychology-i/data-distributions-and-skewness/in-a-right-skewed-income-distribution-the-mean-sits-above-the-median

### 2. House prices, hospital stays and reaction times are all right-skewed. What do these variables have in common that produces the long right tail?

A. They are all measured with instruments that are more accurate at low values than at high ones.
B. A hard floor at zero and no ceiling, so values pile up at the bottom.
C. They are all influenced by many small independent factors, which is what produces a long tail.
D. They are all measured in units that cannot be negative, which is what skewness means.

**Answer:** B. A hard floor at zero and no ceiling, so values pile up at the bottom.

**Why:** The bounded side compresses the values and the unbounded side lets them stretch: nobody responds in 50 milliseconds and an inattentive trial runs to several seconds. Keep it distinct from the opposite shape, many small additive influences produce a *normal* distribution, and skew arises when influences are multiplicative or a boundary is doing the work.

Page: https://tryals.app/practice/psychology-i/data-distributions-and-skewness/house-prices-hospital-stays-and-reaction-times-are-all-right-skewed

### 3. Sort each real-world variable by its likely skew.

**Answer:**

- Right (positive) skew: Personal wealth across a country, Reaction times (cannot go below zero, occasionally very slow)
- Left (negative) skew: Scores on a very easy quiz, Age at death in a country with good healthcare

**Why:** Wealth and reaction times have rare very-high values (right tail); easy-quiz scores and age at death have rare very-low values (left tail). The rare extreme end is the tail, and the tail names the skew.

Page: https://tryals.app/practice/psychology-i/data-distributions-and-skewness/sort-each-real-world-variable-by-its-likely-skew

### 4. A dataset contains one value far from the rest. What is wrong with the common advice to *"remove the outlier and rerun"*?

A. That it decides on the result rather than the reason
B. That outliers cannot be identified reliably without a formal test
C. That removing it reduces the sample size, weakening statistical power
D. Nothing, outliers distort statistics and should always be removed

**Answer:** A. That it decides on the result rather than the reason

**Why:** The question is why the point is there, not how far out it sits. A mistyped decimal should go; a genuinely extreme participant is data. Deciding after seeing the effect on the result is the forking-path problem in miniature, which is why exclusion rules belong in the protocol, written before the data arrive.

Page: https://tryals.app/practice/psychology-i/data-distributions-and-skewness/a-dataset-contains-one-value-far-from-the-rest-what-is-wrong-with

### 5. A charity reports that the mean donation to its campaign was £340. A journalist checks and finds the median donation was £25. Neither figure is wrong. What has almost certainly happened, and which number should the headline use?

A. The median is unreliable for donation data, so £340 is the figure to report.
B. The distribution is left-skewed, with most donations clustered at the high end and a tail of small ones.
C. A few very large gifts have pulled the mean above the bulk.
D. The charity has made an arithmetic error, since the mean and median of the same data cannot differ by that much.

**Answer:** C. A few very large gifts have pulled the mean above the bulk.

**Why:** A mean thirteen times the median is the signature of a strong right skew, and donations have exactly the structure that produces one: a floor at zero, no ceiling, a handful of major gifts. Both figures have legitimate uses, median for the typical supporter, mean for what a donor is worth to the budget. The dishonest move is choosing by which sounds better and not saying which you used.

Page: https://tryals.app/practice/psychology-i/data-distributions-and-skewness/a-charity-reports-that-the-mean-donation-to-its-campaign-was-340-a