Descriptive Statistics

Descriptive statistics help us summarize and organize large sets of data so we can easily understand their main features. We typically look at two main aspects: the center of the data and the spread of the data.

Measures of Center

Measures of center tell us where the middle or typical value of a data set lies.

• Mean (Average): The sum of all values divided by the number of values.
• Median: The middle value when the data is ordered from least to greatest. If there is an even number of values, it is the average of the two middle numbers.
• Mode: The value(s) that appear most frequently.

Example: Find the mean, median, and mode of the data set: $3, 5, 7, 7, 9, 12, 15$.

• Mean: $\frac{3 + 5 + 7 + 7 + 9 + 12 + 15}{7} = \frac{58}{7} \approx 8.29$
• Median: The numbers are already in order. The middle (4th) number is $7$.
• Mode: The number $7$ appears most often.

Measures of Spread

Measures of spread describe how stretched or squeezed the data is.

• Range: The difference between the maximum and minimum values. For the data above, the range is $15 - 3 = 12$.
• Interquartile Range (IQR): The range of the middle 50% of the data. It is calculated as $Q3 - Q1$, where $Q1$ is the median of the lower half and $Q3$ is the median of the upper half.
• Standard Deviation: A measure of how much the individual data values deviate from the mean. A low standard deviation means the data is tightly clustered close to the mean.

Box Plots and Outliers

A box plot (or box-and-whisker plot) is a visual display of the five-number summary:

1. Minimum: The lowest value (excluding outliers).
2. First Quartile (Q1): The median of the lower half of the data.
3. Median (Q2): The middle of the data set.
4. Third Quartile (Q3): The median of the upper half of the data.
5. Maximum: The highest value (excluding outliers).

The "box" is drawn from $Q1$ to $Q3$, representing the middle 50% of the data, with a vertical line inside marking the median. The "whiskers" extend outward to the minimum and maximum values.

Identifying Outliers: An outlier is an unusually high or low value that stands out from the rest of the data. Mathematically, a value is considered an outlier if it is:

• Less than $Q1 - 1.5 \times IQR$
• Greater than $Q3 + 1.5 \times IQR$