# Descriptive Statistics

Mathematics I · Data Analysis and Probability · https://tryals.app/learn/mathematics-i/descriptive-statistics

## Centre & Spread

**Descriptive statistics** condense raw data to reveal its shape by asking two questions: where is the **centre**, and how much **spread** is there?

| Measure | Formula / Definition | Best Used For |
| --- | --- | --- |
| **Mean** | $\bar{x} = \frac{1}{n}\sum x_i$ (average) | Symmetric data |
| **Median** | Middle value of sorted data | Skewed data / outliers |
| **Mode** | Most frequent value | Categorical data |

The **mean** is sensitive to every value, while the **median** is robust against extremes. Always match your summary metric to the shape of your data.

## Dispersion & Pitfalls

Measures of **spread** quantify how much values vary around the centre:

- **Range**: Maximum minus minimum; highly outlier-sensitive.
- **Variance**: $\sigma^2 = \frac{1}{n}\sum (x_i - \bar{x})^2$, the average squared deviation.
- **Standard deviation**: $\sigma = \sqrt{\sigma^2}$, the square root of variance, kept in the **same units** as the data for clear interpretation.

> **Common pitfall:** Using the **mean** for **skewed data**. Extreme tails pull the mean away from the bulk of observations, making the **median** a better "typical" value. Report **standard deviation**, not raw variance.

## Practice questions

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

### 1. For the data 2, 4, 4, 5, 10 the mean is 5 and the median is 4. Change the 10 to 100 and the mean becomes 23 while the median stays at 4. Why does one summary move so much further than the other?

A. Because 100 is an outlier and outliers are automatically excluded from the median calculation.
B. Because the mean is only valid for symmetric data, so its value is meaningless here.
C. Because the median is a more accurate measure of centre, and the mean contains an error that grows with the data.
D. Because the mean uses every value's size and the median only its rank.

**Answer:** D. Because the mean uses every value's size and the median only its rank.

**Why:** The mean sums the values so every magnitude enters, while the median asks only which value sits third, and the fifth being 10 or a million does not change that. Neither behaviour is a defect: mean salary times headcount is the wage bill, and the median is the typical case. Nothing is being excluded; the 100 is counted as one value among five and simply not asked how big it is.

Page: https://tryals.app/practice/mathematics-i/descriptive-statistics/for-the-data-2-4-4-5-10-the-mean-is-5-and-the-median-is-4-change

### 2. Anscombe's quartet is four datasets with identical means, variances, correlation and regression line, and wildly different shapes. What was Anscombe demonstrating?

A. That identical summaries can describe completely different data
B. That summary statistics should be replaced by visual inspection entirely
C. That four is the minimum number of datasets needed to detect a pattern
D. That correlation coefficients are computed incorrectly by most software

**Answer:** A. That identical summaries can describe completely different data

**Why:** Same numbers, four different worlds: a linear relationship, a parabola, an outlier dragging the line, and a single leverage point. The summaries are correct and insufficient, which is why 'plot it first' is methodological advice rather than aesthetic preference.

Page: https://tryals.app/practice/mathematics-i/descriptive-statistics/anscombes-quartet-is-four-datasets-with-identical-means-variances

### 3. Order the steps for computing the standard deviation of a dataset.

**Answer:**

1. Compute the mean of the data
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 to return to the data's own units

**Why:** Squaring before averaging is essential: raw deviations always average to exactly zero, since positive and negative deviations cancel. Squaring removes the cancellation, and the final square root undoes the inflation of the units.

Page: https://tryals.app/practice/mathematics-i/descriptive-statistics/order-the-steps-for-computing-the-standard-deviation-of-a-dataset

### 4. In the dataset 2, 4, 4, 5, 10 the largest value is changed from 10 to 100. Select every statistic that changes as a result.

A. Standard deviation
B. Mean
C. Range
D. Mode
E. Median

**Answer:** A. Standard deviation; B. Mean; C. Range

**Why:** The mean rises from 5 to 23, the range from 8 to 98, and the spread balloons. The median stays 4 (the third of five sorted values, and 100 is still last) and the mode stays 4. This single edit is a compact demonstration of robustness.

Page: https://tryals.app/practice/mathematics-i/descriptive-statistics/in-the-dataset-2-4-4-5-10-the-largest-value-is-changed-from-10-to