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Relate Measures to
Distribution Shape

Learn how the shape of data tells you which measures of center and variability best describe your data set.

6th Grade Math

What Is a Distribution?

A distribution shows how data values are spread out. When you plot your data on a dot plot or histogram, the overall pattern you see is the distribution's shape.

Data Values Symmetric Distribution

Three Distribution Shapes

Symmetric
Data is evenly spread on both sides of the center: a mirror image.
Skewed Left
A long tail stretches to the left (lower values). Most data clusters right.
Skewed Right
A long tail stretches to the right (higher values). Most data clusters left.
Symmetric Skewed Left Skewed Right

Measures of Center

Mean
Add all values and divide by the count. Sensitive to outliers and skew.
Median
The middle value when data is ordered. Resistant to outliers and skew.

Measures of Variability

MAD
Mean Absolute Deviation: average distance from the mean. Works best with symmetric data.
IQR
Interquartile Range: \( Q_3 - Q_1 \). Resistant to outliers: great for skewed data.

The Core Rule

1
Symmetric distribution: Use mean and MAD to summarize the data.
2
Skewed distribution (or outliers): Use median and IQR to summarize the data.

Visualizing Mean vs. Median

\( \text{Symmetric: Mean} \approx \text{Median} \)
Mean/Median
\( \text{Skewed Right: Mean} > \text{Median} \)
Median Mean pulled

Resistant vs. Non-Resistant

Resistant Non-Resistant
Center Median Mean
Variability IQR MAD
Affected by outliers? No Yes
Best for Skewed data Symmetric data

How to Choose the Right Measures

1
Plot or examine the data. Create a dot plot, histogram, or box plot to see the shape.
2
Identify the shape. Is it symmetric, skewed left, or skewed right? Are there outliers?
3
Choose your measures. Symmetric: use mean + MAD. Skewed or outliers: use median + IQR.
4
Consider context. Think about what the data represents: does a typical value or a middle value make more sense?

Example 1: Test Scores

Data: 78, 80, 82, 83, 84, 85, 85, 86, 87, 88

78 80 82 83 84 85 86 87 88 Roughly Symmetric
\( \text{Mean} = 83.8, \text{Median} = 84.5 \)

Example 2: Home Prices

Data (in thousands): 150, 160, 165, 170, 175, 180, 185, 190, 200, 450

450K 150 200 Long tail (outlier)
\( \text{Median} = 177.5\text{K} \)
\( \text{Mean} = 202.5\text{K} \)

Example 3: Context Matters

A teacher records the number of books students read over summer: 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 15

2 3 4 5 6 7 15 Outlier! Skewed Right

Real-World Context Clues

Some real-world data sets are almost always skewed. Knowing the context can help you choose before even graphing.

Often Skewed Right
Income, home prices, wait times, race finish times: a few extreme high values pull the tail right.
Often Symmetric
Heights, test scores, temperature: values cluster around the center with balance on both sides.

How Outliers Shift the Mean

Without outlier: 10, 12, 13, 14, 15
\( \text{Mean} = 12.8, \text{Median} = 13 \)
With outlier: 10, 12, 13, 14, 15, 50 50
\( \text{Median} = 13.5 \)
\( \text{Mean} = 19 \)
pulled right

Common Mistakes

✕ Always Using the Mean
Students default to the mean for every data set, even when outliers or skew make it misleading.
✕ Ignoring the Graph
Jumping straight to calculations without first looking at the distribution shape leads to wrong measure choices.
✕ Mixing Pairs
Pairing median with MAD (or mean with IQR) creates an inconsistent summary: always pair mean + MAD or median + IQR.

Summary Table

Distribution Shape Best Center Best Variability Why?
Symmetric Mean MAD No outliers to distort
Skewed Left Median IQR Tail pulls mean down
Skewed Right Median IQR Tail pulls mean up
Has Outliers Median IQR Outliers inflate mean

Question 1 of 5

A data set of student heights is symmetric with no outliers. Which measures should you use to summarize the data?

Question 2 of 5

A neighborhood records home sale prices: \( 180\text{K}, 190\text{K}, 195\text{K}, 200\text{K}, 210\text{K}, 850\text{K} \). Which measure of center best describes a typical sale price?

Question 3 of 5

In a skewed-right distribution, what is the relationship between the mean and median?

Question 4 of 5

Why is the IQR a better measure of spread than MAD for data with outliers?

Question 5 of 5

A class collected data on the number of pets students own: 0, 1, 1, 1, 2, 2, 2, 2, 3, 3. The distribution is roughly symmetric. A new student who owns 14 pets joins. Which statement is true?