Describe data distributions

Detailed Explanation: Describe data distributions

A data distribution tells how the values in a data set are spread out. To describe it, look for:

• Center: a typical value, such as the mean or median
• Variability: how much the values differ, such as the range or interquartile range
• Shape: whether the data are balanced, skewed, or have a long tail
• Clusters: groups of values close together
• Gaps: intervals with no values
• Outliers: values far from most of the others

Worked example

A teacher records the number of minutes 12 students spend reading:

1. Describe the center

The median is the middle value. There are 12 values, so use the average of the 6th and 7th values:

The typical reading time is about 5 minutes.

The mean is:

The mean is a little larger than the median because the value $(15)$ pulls the mean upward.

2. Describe the variability

The range is the greatest value minus the least value:

So, the data spread across 13 minutes.

The middle half of the data goes from $(3.5)$ to $6$, so the interquartile range is:

This shows that most values are fairly close together, even though the full range is large.

3. Describe the shape, clusters, gaps, and outliers

• There is a cluster from $2$ to $7$, where most values occur.
• There is a gap from $8$ to $(14)$, because no students have times in that interval.
• The value $(15)$ is a possible outlier because it is far from the other values.
• The distribution is skewed right because of the long stretch toward the larger value $(15)$.

A complete description is:

The data are centered near $5$ minutes, with a median of $5$ and a mean of about $(5.4)$. Most values are clustered between $2$ and $7$. The range is $(13)$, but the middle half varies by only $(2.5)$. There is a gap from $8$ to $(14)$, and $(15)$ is a possible outlier, making the distribution skewed right.

Remember that one number does not tell the whole story. The median, range, and shape together give a clearer description of the distribution.