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

Lesson 1: Stem-and-Leaf Plots

In this Grade 6 lesson from Big Ideas Math, Course 1, students learn how to make and interpret stem-and-leaf plots by organizing data values using stems (leading digits) and leaves (remaining digits). The lesson covers constructing single and back-to-back stem-and-leaf plots, writing a key, and analyzing the distribution of a data set. These skills align with Common Core Standard 6.SP.4 from Chapter 10: Data Displays.

Section 1

Stem-and-Leaf Plots

Property

A stem-and-leaf plot is a way to organize numerical data that shows both the shape of the data distribution and the actual data values.
To create a stem-and-leaf plot, split each data value into two parts: the "stem" (usually the tens digit or higher place values) and the "leaf" (usually the ones digit).
List the stems vertically in order, then write the corresponding leaves horizontally next to each stem.
This type of graph allows us to identify clusters (data points together in a group), gaps (intervals without any reported values), peaks (stems where there are more leaves than for nearby stems), and outliers (values that are significantly different from the rest of the data).

Examples

Section 2

Introduction to Measures of Variation

Property

A measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes with a single number how its values vary.
The first, and easiest, measure is the range. This is given by the difference between the highest and lowest data values.
While easy to compute, it only tells us how far apart the extreme values are and gives no indication about the spread of the data values between the two extremes.

Examples

  • A student's test scores are 85, 92, 78, 95, and 88. The highest score is 95 and the lowest is 78. The range is 95−78=1795 - 78 = 17.
  • Two basketball players' points per game are recorded. Player A: {15, 17, 16, 18}. Player B: {5, 10, 25, 30}. Player A's range is 18−15=318 - 15 = 3, while Player B's is 30−5=2530 - 5 = 25, showing Player B's scoring is less consistent.
  • The data sets {2, 8, 8, 9, 12} and {2, 3, 4, 5, 12} both have a range of 10. However, the first set is clustered high while the second is more evenly spread, showing a limitation of using only the range.

Explanation

Measures of variability, like range, tell you about the spread of your data. While the mean or median tells you the center, variability describes if the data points are all clustered together or widely scattered apart.

Section 3

Calculating the Median from a Stem-and-Leaf Plot

Property

The median is the middle value in an ordered data set. To find the median from a stem-and-leaf plot with nn data values (leaves):

  • If nn is odd, the median is the value at the (n+12)th(\frac{n+1}{2})^{th} position.
  • If nn is even, the median is the average of the values at the (n2)th(\frac{n}{2})^{th} and (n2+1)th(\frac{n}{2}+1)^{th} positions.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Stem-and-Leaf Plots

Property

A stem-and-leaf plot is a way to organize numerical data that shows both the shape of the data distribution and the actual data values.
To create a stem-and-leaf plot, split each data value into two parts: the "stem" (usually the tens digit or higher place values) and the "leaf" (usually the ones digit).
List the stems vertically in order, then write the corresponding leaves horizontally next to each stem.
This type of graph allows us to identify clusters (data points together in a group), gaps (intervals without any reported values), peaks (stems where there are more leaves than for nearby stems), and outliers (values that are significantly different from the rest of the data).

Examples

Section 2

Introduction to Measures of Variation

Property

A measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes with a single number how its values vary.
The first, and easiest, measure is the range. This is given by the difference between the highest and lowest data values.
While easy to compute, it only tells us how far apart the extreme values are and gives no indication about the spread of the data values between the two extremes.

Examples

  • A student's test scores are 85, 92, 78, 95, and 88. The highest score is 95 and the lowest is 78. The range is 95−78=1795 - 78 = 17.
  • Two basketball players' points per game are recorded. Player A: {15, 17, 16, 18}. Player B: {5, 10, 25, 30}. Player A's range is 18−15=318 - 15 = 3, while Player B's is 30−5=2530 - 5 = 25, showing Player B's scoring is less consistent.
  • The data sets {2, 8, 8, 9, 12} and {2, 3, 4, 5, 12} both have a range of 10. However, the first set is clustered high while the second is more evenly spread, showing a limitation of using only the range.

Explanation

Measures of variability, like range, tell you about the spread of your data. While the mean or median tells you the center, variability describes if the data points are all clustered together or widely scattered apart.

Section 3

Calculating the Median from a Stem-and-Leaf Plot

Property

The median is the middle value in an ordered data set. To find the median from a stem-and-leaf plot with nn data values (leaves):

  • If nn is odd, the median is the value at the (n+12)th(\frac{n+1}{2})^{th} position.
  • If nn is even, the median is the average of the values at the (n2)th(\frac{n}{2})^{th} and (n2+1)th(\frac{n}{2}+1)^{th} positions.

Examples