Compare data sets using spread

Spread describes how much numerical data values vary around a distribution, complementing measures of center such as the mean or median. A learner compares data sets in the same units by interpreting and calculating range, and when quartiles are provided, the interquartile range, recognizing that a larger spread indicates less consistency and that range can be strongly affected by an extreme value; standard deviation and variance are beyond this scope.

Detailed Explanation: Compare data sets using spread

Spread tells how far apart the values in a data set are. A data set with a larger spread is less consistent because its values vary more.

The two common measures of spread are:

Range: largest value minus smallest value

\text{Range}=\text{maximum}-\text{minimum}

Interquartile range (IQR): third quartile minus first quartile

\text{IQR}=Q_3-Q_1

Example

Two classes recorded the number of minutes students spent reading.

Class A: $8, 9, 10, 10, 11, 12$
Class B: $6, 7, 10, 11, 13, 14$

Both data sets are already in order.

Step 1: Find each range

For Class A:

12-8=4\text{ minutes}

For Class B:

14-6=8\text{ minutes}

Class B has the larger range, so its reading times are more spread out.

Step 2: Find each IQR

For six values, split the data into a lower half and an upper half.

For Class A:

Lower half: $8,9,10$ , so $Q_1=9$
Upper half: $10,11,12$ , so $Q_3=11$

\text{IQR}=11-9=2\text{ minutes}

For Class B:

Lower half: $6,7,10$ , so $Q_1=7$
Upper half: $11,13,14$ , so $Q_3=13$

\text{IQR}=13-7=6\text{ minutes}

Conclusion

Class B has both a larger range ( $8$ minutes compared with $4$ minutes) and a larger IQR ( $6$ minutes compared with $2$ minutes). Therefore, Class B’s reading times are more spread out and less consistent.

Remember that the range uses only the smallest and largest values, so one unusually large or small value can change the range a lot.

Learn by doing: Compare data sets using spread

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Statistics - Quartiles - Data Set (No Outliers) to Interquartile Range

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