11-2 Measures of Spread

Property

A measure of center summarizes a data set with a single "typical" number.

• Mode: The value that occurs most often.
• Median: The exact middle value when the data is ordered from least to greatest. (If there is an even number of values, average the two middle numbers).
• Mean (Average): The sum of all values divided by the total count $n$.

When calculating the mean, you must note whether your data is a Sample (a small surveyed subset) or a Population (the entire group existing in reality).

• Population Mean is represented by the Greek letter $\mu$ (mu).
• Sample Mean is represented by $\bar{x}$ (x-bar).

Examples

• Finding the Median: For the data set {9, 2, 7, 5, 11}, order it first: {2, 5, 7, 9, 11}. The middle value is the 3rd one, so the median is 7.
• Finding the Mean: For five quiz scores {8, 10, 7, 9, 6}, the sum is 40. The mean is $40 \div 5 = 8$.
• Choosing the Best Measure: House prices on a street are 200k, 210k, 225k, 240k, and 950k. The mean is 365k, which is heavily distorted by the one 950k mansion. The median is 225k, which is a much better measure of the "typical" house on this street.

Property

To describe the spread (variability) of data, we calculate specific distances.

• Range: Maximum value minus the Minimum value.
• 5-Number Summary: A breakdown of the data into four equal quarters using five values: Minimum, Q1 (median of the lower half), Median (Q2), Q3 (median of the upper half), and Maximum.
• Interquartile Range (IQR): The difference between the third and first quartiles ($Q_3 - Q_1$). It measures the spread of the middle 50% of the data.

Examples

• Calculating the 5-Number Summary: Given an ordered set {1, 3, 5, 8, 9, 10}.

Min: 1. Max: 10.
Median: The average of 5 and 8 is 6.5.
Q1 (median of lower half {1, 3, 5}): 3.
Q3 (median of upper half {8, 9, 10}): 9.
The summary is (1, 3, 6.5, 9, 10).

• Interpreting IQR: Player A has an IQR of 3 points. Player B has an IQR of 12 points. Player A is the more consistent scorer because their typical, middle 50% of scores vary within a much tighter window.

Explanation

Why do we need the IQR if we already have the total Range? The total Range is easily distorted by just one extreme outlier. The IQR acts as a statistical filter: it chops off the lowest 25% and the highest 25% of the data, focusing strictly on the core middle 50%. A smaller IQR means your data is consistent and tightly clustered; a larger IQR means your data is widely scattered.

Property

A box plot (or box-and-whisker plot) is a visual representation of the 5-Number Summary floating above a number line.

• The "box" spans from Q1 to Q3, visually representing the IQR (the middle 50% of the data).
• A vertical line is drawn inside the box at the Median.
• The "whiskers" extend outward from the box to the Minimum and Maximum values (excluding extreme outliers, which are plotted as individual standalone dots).

Examples

• Identifying Parts: A box plot has whiskers ending at 12 and 48, a box from 20 to 38, and a line inside the box at 27.

Min = 12; Q1 = 20; Median = 27; Q3 = 38; Max = 48.
The IQR is the width of the box: $38 - 20 = 18$.

• The Equal Halves Error: In the plot above, the left side of the box spans 7 units (20 to 27) and the right side spans 11 units (27 to 38). This does NOT mean there is more data on the right. Every section of a box plot ALWAYS contains exactly 25% of the data points. A wider section simply means those data points are more stretched out.

Explanation

A box plot translates a list of numbers into a simple, powerful picture. The most common mistake students make is assuming that a wider section of the box contains more data points. Remember the rule of quarters: the box plot chops your data list into four equally sized groups. If one whisker is very long, it just means that 25% of the data is stretched over a huge distance!