Learn on PengiBig Ideas Math, Course 1Chapter 10: Data Displays

Lesson 3: Shapes of Distributions

In this Grade 6 lesson from Big Ideas Math Course 1, Chapter 10, students learn to identify and describe the shapes of data distributions, including symmetric, skewed left, and skewed right distributions, using dot plots and histograms. Students explore how the position of a distribution's tail determines whether it is skewed and how the shape of a distribution affects the relationship between mean and median. The lesson also guides students in choosing the most appropriate measure of center and variation based on the shape of a given data set.

Section 1

Shapes of Distributions (Symmetric vs. Skewed)

Property

The shape of a data distribution reveals its "personality." When viewing histograms or box plots, we classify the shape into three main categories:

Symmetric: Data is evenly spread around the center. On a box plot, the median is perfectly in the middle of the box, and the whiskers are equal in length.
Skewed Right (Positively Skewed): Most data clusters on the left, with a long "tail" stretching to the right. On a box plot, the median is pushed to the left side of the box ( $Q_1$ ), and the right whisker is much longer.
Skewed Left (Negatively Skewed): Most data clusters on the right, with a long "tail" stretching to the left. On a box plot, the median is pushed to the right side of the box ( $Q_3$ ), and the left whisker is much longer.

Note on Bin Width: When using technology to graph a histogram, choosing the wrong bin width can artificially hide the true shape. Bins that are too wide will make a skewed distribution look deceptively symmetric, while bins that are too narrow will create jagged, fake gaps.

Examples

Symmetric: A dot plot of daily temperatures shows values clustered evenly around 72°F. The left and right sides look like mirror images.
Skewed Right: A histogram of house prices shows most homes cost between $200k–$ 300k (a tall peak on the left), but a few $1M+ mansions create a long tail dragging to the right.
Skewed Left: A box plot of retirement ages has $Q_1 = 58$ , Median = 65, and $Q_3 = 68$ . The left side of the box (58 to 65) is much wider than the right side (65 to 68), and the left whisker stretches far out to early retirees at age 45.

Section 2

Relationship Between Distribution Shape and Measures of Center

Property

The shape of a distribution determines the relationship between the mean and median, which helps us choose the most appropriate measure of center.

Skewed Left: The mean is pulled to the left of the median by extreme low values. The median is a better measure of center because it is not affected by the outliers.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Shapes of Distributions (Symmetric vs. Skewed)

Property

The shape of a data distribution reveals its "personality." When viewing histograms or box plots, we classify the shape into three main categories:

Symmetric: Data is evenly spread around the center. On a box plot, the median is perfectly in the middle of the box, and the whiskers are equal in length.
Skewed Right (Positively Skewed): Most data clusters on the left, with a long "tail" stretching to the right. On a box plot, the median is pushed to the left side of the box ( $Q_1$ ), and the right whisker is much longer.
Skewed Left (Negatively Skewed): Most data clusters on the right, with a long "tail" stretching to the left. On a box plot, the median is pushed to the right side of the box ( $Q_3$ ), and the left whisker is much longer.

Examples

Symmetric: A dot plot of daily temperatures shows values clustered evenly around 72°F. The left and right sides look like mirror images.
Skewed Right: A histogram of house prices shows most homes cost between $200k–$ 300k (a tall peak on the left), but a few $1M+ mansions create a long tail dragging to the right.
Skewed Left: A box plot of retirement ages has $Q_1 = 58$ , Median = 65, and $Q_3 = 68$ . The left side of the box (58 to 65) is much wider than the right side (65 to 68), and the left whisker stretches far out to early retirees at age 45.

Section 2

Relationship Between Distribution Shape and Measures of Center

Property

The shape of a distribution determines the relationship between the mean and median, which helps us choose the most appropriate measure of center.

Skewed Left: The mean is pulled to the left of the median by extreme low values. The median is a better measure of center because it is not affected by the outliers.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Shapes of Distributions (Symmetric vs. Skewed)

Property

The shape of a data distribution reveals its "personality." When viewing histograms or box plots, we classify the shape into three main categories:

Symmetric: Data is evenly spread around the center. On a box plot, the median is perfectly in the middle of the box, and the whiskers are equal in length.
Skewed Right (Positively Skewed): Most data clusters on the left, with a long "tail" stretching to the right. On a box plot, the median is pushed to the left side of the box ( $Q_1$ ), and the right whisker is much longer.
Skewed Left (Negatively Skewed): Most data clusters on the right, with a long "tail" stretching to the left. On a box plot, the median is pushed to the right side of the box ( $Q_3$ ), and the left whisker is much longer.

Examples

Symmetric: A dot plot of daily temperatures shows values clustered evenly around 72°F. The left and right sides look like mirror images.
Skewed Right: A histogram of house prices shows most homes cost between $200k–$ 300k (a tall peak on the left), but a few $1M+ mansions create a long tail dragging to the right.
Skewed Left: A box plot of retirement ages has $Q_1 = 58$ , Median = 65, and $Q_3 = 68$ . The left side of the box (58 to 65) is much wider than the right side (65 to 68), and the left whisker stretches far out to early retirees at age 45.

Section 2

Relationship Between Distribution Shape and Measures of Center

Property

The shape of a distribution determines the relationship between the mean and median, which helps us choose the most appropriate measure of center.

Skewed Left: The mean is pulled to the left of the median by extreme low values. The median is a better measure of center because it is not affected by the outliers.