The interquartile range (IQR) represents the middle 50% of the data.

• This is another measure of variability!

• Recall that the median cut the data in half: 50% of the data is below and 50% is above the median.
• This is also called the 50th percentile.
• The \(p\)th percentile is the value for which \(p\)% of the data is below it.

To get the middle 50%, we will split the data into four parts:

The 25th and 75th percentiles, along with the median, divide the data into four parts.

We call these three measurements the quartiles:

• Q1, the first quartile, is the median of the lower 50% of the data.
• Q2, the second quartile, is the median.
• Q3, the third quartile, is the median of the upper 50% of the data.

Consider {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

• Cutting the data in half: {1, 2, 3, 4, 5 | 6, 7, 8, 9, 10}
• So the median (Q2) is \(\frac{5+6}{2}=5.5\).
• Q1 is the median of {1, 2, 3, 4, 5}, or 3
• Q3 is the median of {6, 7, 8, 9, 10}, or 8

Note: this is a “quick and dirty” way of finding quartiles. A computer will give a more exact result.

Then the interquartile range is \[ \text{IQR} = \text{Q3}-\text{Q1} \]

• The IQR is resistant to extreme values.

• mean and standard deviation when the data are symmetric
• median and IQR when the data are skewed

Box plots summarize the data with 5 statistics plus extreme observations:

1. Draw the vertical axis to include all possible values in the data.
2. Draw a horizontal line at the median, at \(\text{Q1}\), and at \(\text{Q3}\). Use these to form a box.
3. Draw the whiskers. The whiskers’ upper limit is \(\text{Q3}+1.5\times\text{IQR}\) and the lower limit is \(\text{Q1}-1.5\times\text{IQR}\). The actual whiskers are then drawn at the next closest data points within the limits.
4. Any points outside the whisker limits are included as individual points. These are potential outliers.

We won’t draw box plots by hand, but understanding how they are drawn will help us understand how to interpret them!

(Potential) outliers can help us…

• examine skew (outliers in the negative direction suggest left skew; outliers in the positive direction suggest right skew).
• identify issues with data collection or entry, especially if the value of the outliers doesn’t make sense.

• We’ve thought about calculating various descriptive statistics from a sample.
• Our long-term goal is to estimate descriptive information about a population.
• At the population level, these values are called parameters.

• When we find a measure of center, spread, or position, we use a sample to calculate a single value.
• These single values are called point estimates.
  • They are used to estimate the corresponding population parameter.