Interquartile Range

Spread of the Middle Half of the Data

The mean describes a data set's center, but not its spread. Two groups can have the same average age even when their ages are distributed very differently.

The interquartile range ($\mathrm{IQR}$) measures that spread by focusing on the middle $50\%$ of the data after sorting.

Why focus on the middle? Very small or large values can dominate a measure such as the full range. Some may be classified as outliers under a stated rule or through domain judgment. The $\mathrm{IQR}$ is less sensitive to those extremes.

The formula is:

• $Q_3$ is the upper quartile (the value marking the bottom $75\%$ of the data).
• $Q_1$ is the lower quartile (the value marking the bottom $25\%$ of the data).

So, the $\mathrm{IQR}$ is the difference between the upper and lower quartiles.

Comparing Age Spreads

Consider two groups with the same mean but very different spreads.

The table lists $12 \text{ people}$ in each group:

We will compare their means, quartiles, ranges, and interquartile ranges.

Calculating the Mean and Quartiles

1. Mean (Average):

   If you calculate the average age for both groups, the result is exactly the same, $16 \text{ years}$ old.

   Group 1:

   Add all twelve values first, then divide by twelve:

   $$\begin{aligned} &13+14+15+15 \\ &\quad{}+16+16+17+17 \\ &\quad{}+17+17+17+18 \\ &=192 \end{aligned}$$

   $$\frac{192}{12}=16$$

   Group 2:

   Add all twelve values first, then divide their total by twelve.

   $$\begin{aligned}&1+3+4+5\\&{}+7+8+12+27\\&{}+28+29+32+36=192\end{aligned}$$

   $$\frac{192}{12}=16$$

2. Quartiles ($Q_1$ and $Q_3$):

   After sorting the data (as in the table above), we find the positions of the quartiles.

   Group One ($n=12$): The position of $Q_1$ is at data point $\frac{1}{4}(12+1) = 3.25$. This means $Q_1$ is between data points $3$ ($15$) and $4$ ($15$). Since both values are the same, then

   $$Q_1 = 15$$

   The position of $Q_3$ is at data point $\frac{3}{4}(12+1) = 9.75$. This means $Q_3$ is between data points $9$ ($17$) and $10$ ($17$). Since both values are the same, then

   $$Q_3 = 17$$

   Group Two ($n=12$): The position of $Q_1$ is $3.25$. It lies one quarter of the way from value $3$ ($4$) to value $4$ ($5$), so we interpolate with weight $0.25$:

   $$Q_1 = 4 + 0.25(5-4) = 4.25$$

   The position of $Q_3$ is $9.75$. It lies three quarters of the way from value $9$ ($28$) to value $10$ ($29$), so the interpolation weight is $0.75$:

   $$Q_3 = 28 + 0.75(29-28) = 28.75$$

Calculating the Range and Interquartile Range

Next, calculate the measures of spread.

1. Range:

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

   • Range Group One is $18 - 13 = 5$

   • Range Group Two is $36 - 1 = 35$

     The full ranges already show that Group Two is much more spread out at the extremes.

2. Interquartile Range ($\mathrm{IQR}$):

   $\mathrm{IQR} = Q_3 - Q_1$

   • $\mathrm{IQR}$ Group One is $17 - 15 = 2$

   • $\mathrm{IQR}$ Group Two is $28.75 - 4.25 = 24.5$

     The middle halves of the two groups also have very different spreads.

Interpreting the Results

The results can now be compared in one table:

• Both groups have the same mean age, $16$.
• Group One has a range of $5$ and an $\mathrm{IQR}$ of $2$. Its middle $50\%$ lies between $Q_1=15$ and $Q_3=17$, so those ages span only $2\text{ years}$.
• Group Two has a range of $35$ and an $\mathrm{IQR}$ of $24.5$. Its middle $50\%$ lies between $Q_1=4.25$ and $Q_3=28.75$, a span of $24.5\text{ years}$.

Two datasets can have the same mean but different spreads. The $\mathrm{IQR}$ measures the width of the middle half of the data and is resistant to extreme values because it does not use the outer quarters.