Range, IQR, and Variance

Comparing Measures of Spread

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Measures of spread describe how much a dataset varies, not just where its center lies. Range, interquartile range, and variance each summarize spread in a different way, so they help students compare datasets more completely. Two sets of numbers can have the same mean or median but very different variability. Understanding spread is essential in science, economics, psychology, and any field that interprets data.

Range looks at the full distance from the smallest value to the largest, so it is simple but sensitive to extreme values. IQR focuses on the middle 50 percent of the data, making it more resistant to outliers and useful with skewed distributions. Variance measures how far values tend to fall from the mean by averaging squared deviations. Together, these tools reveal whether data are tightly clustered, broadly scattered, or affected by unusual observations.

Key Facts

Range = $\text{maximum} - \text{minimum}$
IQR = $Q_3 - Q_1$
Variance for a population: $\sigma^2 = \frac{\sum (x - \mu)^2}{N}$
Variance for a sample: $s^2 = \frac{\sum (x - \bar{x})^2}{n - 1}$
A larger variance means data values are, on average, farther from the mean
Range uses only two values, IQR uses the middle half of the data, and variance uses every data value

Vocabulary

Range: The range is the difference between the largest and smallest values in a dataset.
Interquartile Range: The interquartile range is the difference between the third quartile and the first quartile, showing the spread of the middle 50 percent of the data.
Variance: Variance is a measure of spread found by averaging the squared distances of data values from the mean.
Quartile: A quartile is a value that divides ordered data into four equal parts.
Outlier: An outlier is a data value that is much larger or smaller than most of the other values.

Common Mistakes to Avoid

Using range to describe typical spread in data with outliers, because one extreme value can change the range a lot and give a misleading picture of most of the dataset.
Finding IQR without ordering the data first, because quartiles must be located from the data in sorted order.
Computing variance by averaging plain deviations from the $\text{mean}$ , because positive and negative deviations cancel and do not measure spread correctly.
Using $n$ instead of $n - 1$ for sample variance, because dividing by $n - 1$ gives the standard unbiased sample-based formula.

Practice Questions

1 For the dataset $2, 4, 6, 8, 10$ , find the range, the mean, and the population variance.
2 For the ordered dataset $1, 3, 4, 6, 8, 9, 11, 15$ , find $Q_1$ , $Q_3$ , and the $IQR$ .
3 Two datasets have the same median, but one has a much larger $IQR$ and $\text{variance}$ . Explain what this tells you about how the data are distributed.