Measures of Center and Variability

When analyzing a set of data, two of the most important things to understand are where the data is centered and how spread out it is. We use measures of center to find a typical value, and measures of variability (or spread) to see how consistent the data is.

Measures of Center: Mean and Median

Measures of center describe the "middle" or typical value of a data set.

• Mean (Average): Found by adding all the numbers together and dividing by the total number of values.
• Median: The middle number when the data is ordered from least to greatest. If there is an even number of values, the median is the mean of the two middle numbers.

Example: Find the mean and median of $\{2, 5, 7, 8, 13\}$.

1. Mean: $\text{Mean} = \frac{2 + 5 + 7 + 8 + 13}{5} = \frac{35}{5} = 7$
2. Median: The numbers are already in order. The middle number in this set of 5 values is the 3rd one. $\text{Median} = 7$

Measures of Variability: MAD and IQR

Measures of variability describe how spread out or clustered the data points are.

• Mean Absolute Deviation (MAD): The average distance of each data point from the mean. A higher MAD means the data is more spread out.
• Interquartile Range (IQR): The range of the middle 50% of the data. It is calculated by subtracting the first quartile ($Q_1$) from the third quartile ($Q_3$). $IQR = Q_3 - Q_1$

Example: Calculate the MAD of $\{3, 5, 7, 9, 11\}$.

1. Find the mean: $\text{Mean} = \frac{3 + 5 + 7 + 9 + 11}{5} = \frac{35}{5} = 7$
2. Find the absolute distance of each number from the mean:
   • $|3 - 7| = 4$
   • $|5 - 7| = 2$
   • $|7 - 7| = 0$
   • $|9 - 7| = 2$
   • $|11 - 7| = 4$
3. Find the mean of these distances: $\text{MAD} = \frac{4 + 2 + 0 + 2 + 4}{5} = \frac{12}{5} = 2.4$ The MAD is $2.4$, meaning on average, each data point is $2.4$ units away from the mean.

Summary

• Use Mean and Median to answer: "What is a typical value in this group?"
• Use MAD and IQR to answer: "How much do the values vary from that typical value?"

If two classes have the same mean test score, but Class A has a MAD of $2$ and Class B has a MAD of $10$, Class A's scores were much more consistent, while Class B had a wider mix of high and low scores.