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Measures of
Center & Variation

Learn how a single number can summarize a whole data set, and how another number tells you how spread out the values are.

6th Grade Math

What Is a Measure of Center?

A measure of center is a single number that represents the "typical" or "middle" value in a data set. Instead of listing every number, you can describe the whole group with one value.

Mean
Add all values, then divide by how many there are. The "average."
Median
The middle value when data is ordered from least to greatest.

What Is a Measure of Variation?

A measure of variation (also called "spread") is a single number that tells you how much the values in a data set differ from each other. A small variation means values are close together; a large variation means they are spread out.

Range
The difference between the greatest and least values.
MAD
Mean Absolute Deviation: the average distance each value is from the mean.

Seeing Center vs Variation

Data Set A 1 2 3 4 5 6 7 8 9
\( \text{Mean} = 5 \)
 Low Variation Data Set B 1 2 3 4 5 6 7 8 9
\( \text{Mean} = 5 \)
 High Variation Both sets have the same center, but different spread!

Computing the Mean

The mean is the sum of all data values divided by the number of values. It answers: "If we shared equally, what would each value be?"

1
Add every value in the data set together.
2
Count how many values there are.
3
Divide the sum by the count.

Finding the Median

The median is the middle value once you line up your data from least to greatest. If there is an even number of values, the median is the mean of the two middle values.

1
Order the values from smallest to largest.
2
If the count is odd, the median is the single middle value.
3
If the count is even, average the two middle values: \(\text{Median} = \dfrac{\text{middle}_1 + \text{middle}_2}{2}\).

Range and MAD: Measuring Spread

Range
\(\text{Range} = \text{max} - \text{min}\)

Quick to calculate, but only uses two values. One outlier can make it huge.
Mean Absolute Deviation
\(\text{MAD} = \dfrac{\sum |x_i - \text{mean}|}{n}\)

Uses every value. Tells the average distance from the mean.

Center vs Variation: The Key Idea

Measure of Center Measure of Variation
Purpose Summarizes with a "typical" value Describes how values differ
Question it answers "What is a representative value?" "How spread out are the values?"
Examples Mean, Median Range, MAD
If all values are the same Equals that value Equals \(0\)

Step-by-Step: Analyze Any Data Set

1
Order the data from least to greatest.
2
Find the mean: add all values and divide by the count.
3
Find the median: identify the middle value (or average the two middle values).
4
Find the range: subtract the smallest value from the largest.
5
Find the MAD: compute each distance from the mean, then average those distances.
6
Interpret: explain what the center and variation tell you about the data.

Example 1: Finding the Mean

Five students scored the following on a quiz: 6, 8, 7, 9, 10. What is the mean score?

5 6 7 8 9 10
\( \text{Mean} = 8 \)

Example 2: Finding the Median

Data set: 3, 7, 2, 9, 5, 1. Find the median.

1
Order: 1, 2, 3, 5, 7, 9
2
Count: 6 values (even), so average the 3rd and 4th values.
3
Middle values: 3 and 5.

Example 3: Range and MAD

Data: 4, 6, 5, 7, 8. The mean is \(\dfrac{4+6+5+7+8}{5} = 6\). Find the range and MAD.

Range
\(8 - 4 = \) \(4\)
MAD Calculation
Distances from mean 6:
\(|4-6| = 2,\; |6-6| = 0,\; |5-6| = 1,\; |7-6| = 1,\; |8-6| = 2\)

Example 4: Comparing Two Classes

Class A test scores: 80, 82, 78, 81, 79. Class B: 60, 100, 70, 90, 80. Compare center and variation.

Class A Class B
Mean 80 80
Range 4 40
MAD 1.2 12

Why Both Measures Matter

City A (Steady) 64 67 70 73 76 79
\( \text{Mean} = 70 \)
\( \text{MAD} = 1.2 \)
 City B (Extreme) 50 58 66 74 82 90
\( \text{Mean} = 70 \)
\( \text{MAD} = 12 \)
 Same center, very different daily life!

City A's temperatures barely change day to day. City B swings wildly. The mean alone would make them look identical.

Common Mistakes to Avoid

✕ Forgetting to Order Data
You must sort values before finding the median. An unsorted list gives the wrong middle value.
✕ Confusing Mean and Median
The mean is computed by adding and dividing. The median is found by position. They are not the same number.
✕ Forgetting Absolute Value in MAD
Distances from the mean must be positive. Without absolute value, negatives cancel and you get the wrong answer.

Summary: All Four Measures

Measure Type How to Find It Sensitive to Outliers?
Mean Center \( \text{Sum} \div \text{count} \) Yes
Median Center Middle value No
Range Variation \( \text{Max} - \text{Min} \) Yes
MAD Variation Avg distance from mean Somewhat

Question 1 of 5

The data set is 3, 5, 7, 9, 11. What is the mean?

Question 2 of 5

For the data 12, 5, 8, 14, 9, 6, what is the median?

Question 3 of 5

A data set has values 20, 25, 30, 35, 40. What is the range?

Question 4 of 5

Which measure tells you how spread out the values in a data set are?

Question 5 of 5

The data set 10, 10, 10, 10, 10 has a MAD of: