# Descriptive Statistics

Psychology I · Statistics · https://tryals.app/learn/psychology-i/descriptive-statistics

## Finding the Center

**Descriptive statistics** summarize data without drawing broader inferential conclusions. **Measures of central tendency** describe a data set's typical value.

| Measure | Definition | Best used when... |
| --- | --- | --- |
| **Mean** | Sum divided by count | Data is symmetric |
| **Median** | Middle value when ordered | Data is skewed or has outliers |
| **Mode** | Most frequent value | Data is categorical |

> **Common pitfall:** Always defaulting to the **mean** without checking for outliers. One extreme value can drag the mean far from what is typical; the **median** is much more robust.

## Measuring Spread

**Measures of variability** describe how spread out a data set is around its center. The **range** is the highest value minus lowest, but it is sensitive to just two extreme points.

**Variance** measures the average squared distance of each point from the mean. The **standard deviation (SD)** is the square root of variance, expressing spread in original units:

$$SD = \sqrt{\text{variance}}$$

| Measure | What it captures |
| --- | --- |
| **Range** | Total spread (high minus low) |
| **Variance** | Average squared distance |
| **Standard deviation** | Spread in original units |

A small **SD** means data clusters tightly; a large **SD** means wide spread.

## Practice questions

4 of this lesson's 9 practice questions, with answers. The full set is in the app.

### 1. You add one extreme outlier to a data set. Sort each statistic by how much that single outlier moves it.

**Answer:**

- Strongly affected by the outlier: The mean, The range, The standard deviation
- Barely affected by the outlier: The median, The mode

**Why:** The mean, range, and standard deviation all incorporate the outlier's raw magnitude, so they shift strongly; the median (a position) and mode (a frequency) barely move. This robustness split is the whole reason we ever prefer the median.

Page: https://tryals.app/practice/psychology-i/descriptive-statistics/you-add-one-extreme-outlier-to-a-data-set-sort-each-statistic-by-how

### 2. Two classes both average 70%. One has a standard deviation of 3, the other of 20. What does a teacher learn from the second number that the mean concealed?

A. Whether 70 describes the class or nobody in it
B. That the second class contains an error, since scores should cluster around the mean
C. That the second class is larger, which widens the spread
D. Nothing about teaching, spread is a statistical property without practical meaning

**Answer:** A. Whether 70 describes the class or nobody in it

**Why:** Same centre, different classroom. A spread of 3 means one lesson fits everyone; a spread of 20 means the average describes a gap rather than a person, and half the room needs something else. Reporting a mean with no spread hides exactly the thing that decides what to do.

Page: https://tryals.app/practice/psychology-i/descriptive-statistics/two-classes-both-average-70-one-has-a-standard-deviation-of-3-the

### 3. Put the steps for computing a standard deviation in the correct order.

**Answer:**

1. Compute the mean of the data set.
2. Subtract the mean from each value to get its deviation.
3. Square each deviation.
4. Average the squared deviations to get the variance.
5. Take the square root of the variance to get the standard deviation.

**Why:** Mean first, then deviations, then square them, average to variance, and finally square-root back to the standard deviation. Squaring before averaging is what prevents positive and negative deviations from cancelling to zero.

Page: https://tryals.app/practice/psychology-i/descriptive-statistics/put-the-steps-for-computing-a-standard-deviation-in-the-correct-order

### 4. Seven salaries are 32, 35, 38, 40, 42, 45 and 220 thousand. The mean is about 64; the median is 40. Why does the median describe this workforce better, when the mean uses all seven numbers and the median effectively uses one?

A. Because the mean is only valid for data sets larger than seven observations.
B. Because using every value means being moved by every value.
C. Because the median is always the more accurate measure of central tendency, whatever the distribution.
D. Because the 220 is certainly a data-entry error, and the median discards errors automatically.

**Answer:** B. Because using every value means being moved by every value.

**Why:** Using every value means being moved by every value: the 220 pulls the mean 24 thousand above where six of the seven people are, so the 'average salary' describes nobody in the room. Do not overcorrect, for symmetric data the mean is more efficient. Check the shape first, and distrust any report of an average that does not say which one it means.

Page: https://tryals.app/practice/psychology-i/descriptive-statistics/seven-salaries-are-32-35-38-40-42-45-and-220-thousand-the-mean