Measures of Variability: Range and Interquartile Range (IQR) - Seventh Grade Mathematics

Measures of variability quantify the spread, dispersion, and consistency of data, dividing distributions into quartiles and box plots to isolate the middle fifty percent.

The Importance of Measuring Dispersion

Knowing only the center of a dataset provides an incomplete mathematical picture. Two datasets can share the exact same mean yet possess radically different levels of consistency.

               TWO DATASETS WITH IDENTICAL MEAN (50)
   Dataset 1:  49, 50, 50, 51   --> Mean = 50  (High consistency / low spread)
   Dataset 2:   0, 20, 80, 100  --> Mean = 50  (Low consistency / high spread)

Measures of variability quantify how spread out data points are around the center, revealing predictability and risk.

The Range

The simplest measure of variability is the range, defined as the difference between the single greatest value (maximum) and the single smallest value (minimum) in the dataset.

               THE RANGE FORMULA
              Range = Maximum Value - Minimum Value

If a basketball player scores between eight and thirty-four points in various games, the range is thirty-four minus eight, which equals twenty-six points. While simple to calculate, the range has a severe limitation: because it depends entirely on the two extreme endpoints, a single outlier can distort the reported spread dramatically.

Quartiles and the Five-Number Summary

To understand the internal distribution of data without distortion from outliers, statisticians divide an ordered dataset into four equal segments called quartiles.

               THE THREE QUARTILE CUTOFFS
   Q1 (First Quartile):  Median of lower half (25th percentile mark)
   Q2 (Second Quartile): Median of entire dataset (50th percentile mark)
   Q3 (Third Quartile):  Median of upper half (75th percentile mark)

Together with the minimum and maximum, the quartiles form the five-number summary:

+-----------------------+--------------------------------------------------+
| Component             | Statistical Meaning                              |
+-----------------------+--------------------------------------------------+
| Minimum               | The smallest value in the dataset                |
| First Quartile (Q1)   | Cuts off bottom 25% from upper 75%               |
| Median (Q2)           | Divides dataset into two equal halves (50%)      |
| Third Quartile (Q3)   | Cuts off bottom 75% from upper 25%               |
| Maximum               | The largest value in the dataset                 |
+-----------------------+--------------------------------------------------+

When finding Q1 and Q3, if the total dataset count n is odd, the median Q2 is excluded from both halves when determining the lower and upper medians.

The Interquartile Range (IQR)

The Interquartile Range (IQR) measures the spread of the middle fifty percent of the data, calculated by subtracting the first quartile from the third quartile.

               THE INTERQUARTILE RANGE FORMULA
                           IQR = Q3 - Q1

Because the IQR focuses exclusively on the central fifty percent, it completely discards extreme outliers and tails, providing a highly resistant and reliable measure of dispersion.

Constructing Box-and-Whisker Plots

A box-and-whisker plot (or box plot) provides a graphical summary of the five-number summary along a calibrated number line.

               ANATOMY OF A BOX-AND-WHISKER PLOT
                 |--------[     |        ]--------|
                Min      Q1    Median   Q3       Max
                 |<--25%->|<---25%--->|<-25%-->|<-25%-->|
                          |<------ IQR ------->|
  <----+----+----+----+----+----+----+----+----+----+----+---->
      10   20   30   40   50   60   70   80   90  100

To construct a box plot:
1. Draw a calibrated numerical scale below.

2. Plot vertical line segments at Q1, Median, and Q3, and connect them to form a central rectangular box.

3. Plot endpoints at the Minimum and Maximum.

4. Draw horizontal whisker lines extending from the box out to the Minimum and Maximum.
The length of the box represents the IQR, while the total span from whisker tip to whisker tip represents the range.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. A dataset records the daily high temperatures in degrees Fahrenheit over ten days: sixty-two, sixty-five, sixty-eight, seventy, seventy-two, seventy-four, seventy-six, eighty, eighty-four, and ninety. Determine the range of temperatures.

Exercise 2. For the temperature dataset in Exercise 1, find the five-number summary (Minimum, Q1, Median, Q3, Maximum) and compute the Interquartile Range (IQR).

Exercise 3. A student's test scores are: fifty-five, seventy-two, seventy-six, eighty, eighty-two, eighty-six, eighty-eight, ninety, and ninety-four. Identify the median, the first quartile Q1, and the third quartile Q3.

Exercise 4. Explain why the Interquartile Range is considered a more reliable measure of spread than the range for a salary dataset containing several multi-million-dollar executive bonuses.

Solutions

Solution to Exercise 1. The daily temperatures are already sorted from least to greatest: sixty-two, sixty-five, sixty-eight, seventy, seventy-two, seventy-four, seventy-six, eighty, eighty-four, ninety. The minimum temperature is sixty-two, and the maximum temperature is ninety. Calculate the range: ninety minus sixty-two equals twenty-eight degrees Fahrenheit.

Solution to Exercise 2. The dataset has ten values. The minimum is sixty-two, and the maximum is ninety. The median Q2 is the average of the fifth and sixth values: seventy-two plus seventy-four equals one hundred forty-six; one hundred forty-six divided by two equals seventy-three. The lower half of the data consists of the first five values: sixty-two, sixty-five, sixty-eight, seventy, seventy-two. The median of this lower half is the middle value, sixty-eight, so Q1 equals sixty-eight. The upper half consists of the top five values: seventy-four, seventy-six, eighty, eighty-four, ninety. The median of the upper half is eighty, so Q3 equals eighty. The five-number summary is: Minimum 62, Q1 68, Median 73, Q3 80, Maximum 90. The Interquartile Range is Q3 minus Q1: eighty minus sixty-eight equals twelve degrees Fahrenheit.

Solution to Exercise 3. The dataset has nine values (an odd count): fifty-five, seventy-two, seventy-six, eighty, eighty-two, eighty-six, eighty-eight, ninety, ninety-four. The median Q2 is the fifth value: eighty-two. Exclude eighty-two to form the lower and upper halves. The lower half contains four values: fifty-five, seventy-two, seventy-six, eighty. The median Q1 is the average of seventy-two and seventy-six: seventy-two plus seventy-six equals one hundred forty-eight; one hundred forty-eight divided by two equals seventy-four. The upper half contains four values: eighty-six, eighty-eight, ninety, ninety-four. The median Q3 is the average of eighty-eight and ninety: eighty-eight plus ninety equals one hundred seventy-eight; one hundred seventy-eight divided by two equals eighty-nine. Thus: Q1 = 74, Median = 82, Q3 = 89.

Solution to Exercise 4. The range is calculated as maximum salary minus minimum salary. If several executives earn multi-million-dollar bonuses, the maximum jumps dramatically, stretching the range to millions of dollars. This creates the misleading impression that all employee salaries vary wildly by millions of dollars, when in fact ninety-five percent of workers may earn within five thousand dollars of one another. The Interquartile Range measures the middle fifty percent of salaries between Q1 and Q3, completely ignoring the extreme executive bonuses at the top. The IQR faithfully reflects the true salary variation experienced by typical workers.