Measures of Center
& Variability

Master the mean, median, IQR, and MAD: the essential tools for describing what is typical in a data set and how spread out the values are.

6th Grade Math

What Are Measures of Center?

A measure of center is a single number that represents what is typical in a data set. It tells you where the "middle" of the data lives.

Mean

The average: add all values and divide by how many there are.

Median

The middle value when data is ordered from least to greatest.

What Are Measures of Variability?

A measure of variability (or spread) tells you how spread out the data values are. Two data sets can share the same center but look very different.

IQR

Interquartile Range: the spread of the middle 50% of data. \( \text{IQR} = Q_3 - Q_1 \)

MAD

Mean Absolute Deviation: the average distance each value is from the mean.

Visualizing Center and Spread

The Mean: Adding & Dividing

1

Add all the data values together.

2

Count how many values there are (\( n \)).

3

Divide the sum by \( n \).

The Median: The Middle Value

To find the median, arrange all data from least to greatest, then locate the middle value.

Odd Count

The median is the single middle number. For \( n \) values, it is in position \( \frac{n+1}{2} \).

Even Count

The median is the average of the two middle numbers.

Mean vs. Median

	Mean	Median
What it measures	Average of all values	Middle value
Affected by outliers?	Yes	No
Best for	Symmetric data	Skewed data
Uses all values?	Yes	Only position matters

Interquartile Range (IQR)

Mean Absolute Deviation (MAD)

1

Find the mean \( \bar{x} \) of the data set.

2

Subtract the mean from each value and take the absolute value: \( |x_i - \bar{x}| \).

3

Add all the absolute deviations together.

4

Divide by \( n \) to find the average deviation.

How to Describe a Data Set

1

Order the data from least to greatest.

2

Find the center: calculate the mean and/or median.

3

Find the spread: calculate the IQR and/or MAD.

4

Look for patterns: is the data symmetric, skewed, or clustered?

5

Spot outliers: any values far from the rest? Note how they affect center and spread.

Example 1: Mean & Median

Data: Test scores for 7 students: 72, 85, 90, 88, 76, 95, 82

1

Order: 72, 76, 82, 85, 88, 90, 95

2

Mean: \( \frac{72+76+82+85+88+90+95}{7} = \frac{588}{7} = \) 84

3

Median: The 4th value in the ordered list = 85

Example 2: Finding the IQR

Ordered data: 72, 76, 82, 85, 88, 90, 95

1

Lower half (below median): 72, 76, 82 → \( Q_1 = \) 76

2

Upper half (above median): 88, 90, 95 → \( Q_3 = \) 90

3

IQR: \( Q_3 - Q_1 = 90 - 76 = \) 14

Example 3: Calculating the MAD

Data: 3, 5, 7, 9, 11 → Mean: \( \bar{x} = \frac{35}{5} = 7 \)

\( x_i \)	\( x_i - \bar{x} \)	\( \|x_i - \bar{x}\| \)
3	\( 3 - 7 = -4 \)	4
5	\( 5 - 7 = -2 \)	2
7	\( 7 - 7 = 0 \)	0
9	\( 9 - 7 = 2 \)	2
11	\( 11 - 7 = 4 \)	4

Spotting Outliers & Patterns

A striking deviation (outlier) is a data value far from the rest. Outliers strongly affect the mean but barely change the median.

Common Mistakes to Avoid

✕ Forgetting to Order

Finding the median without first arranging data from least to greatest gives the wrong middle value.

✕ Wrong Divisor for Mean

For the mean, divide by the total count of values, not by the largest value or some other number.

✕ Skipping Absolute Value

In MAD, negative deviations must become positive. Without absolute value, they cancel to zero.

Summary: All Four Measures

Measure	Type	What It Tells You	Resists Outliers?
Mean	Center	Average value	No
Median	Center	Middle value	Yes
IQR	Variability	Spread of middle 50%	Yes
MAD	Variability	Avg distance from mean	No

Question 1 of 4

Find the mean of the data set: 4, 8, 6, 10, 12

Add all values: \( 4 + 8 + 6 + 10 + 12 = 40 \). Divide by 5: \( \frac{40}{5} = 8 \). The mean is 8.

Question 2 of 4

Find the median of: 3, 7, 1, 9, 5, 11

First, order the data: 1, 3, 5, 7, 9, 11. With 6 values (even count), the median is the average of the 3rd and 4th values: \( \frac{5 + 7}{2} = 6 \).

Question 3 of 4

For the ordered data set 2, 5, 8, 11, 13, 15, 22, what is the IQR?

The median is 11 (4th of 7 values). Lower half: 2, 5, 8, so \( Q_1 = 5 \). Upper half: 13, 15, 22, so \( Q_3 = 15 \). \( \text{IQR} = 15 - 5 = 10 \).

Question 4 of 4

A data set has values: 10, 10, 10, 10, 50. Which measure of center better represents the typical value?

The mean is \( \frac{90}{5} = 18 \), but 4 out of 5 values are 10. The median is 10, which is much more typical. When there is an outlier, the median is the better choice.

Measures of Center& Variability

What Are Measures of Center?

What Are Measures of Variability?

Visualizing Center and Spread

The Mean: Adding & Dividing

The Median: The Middle Value

Mean vs. Median

Interquartile Range (IQR)

Mean Absolute Deviation (MAD)

How to Describe a Data Set

Example 1: Mean & Median

Example 2: Finding the IQR

Example 3: Calculating the MAD

Spotting Outliers & Patterns

Common Mistakes to Avoid

Summary: All Four Measures

Question 1 of 4

Question 2 of 4

Question 3 of 4

Question 4 of 4

Measures of Center
& Variability