Explain How Two Samples Can Have the Same Mean but Different Standard Deviations.

Understanding the Standard Difference

Alignments to Content Standards: Southward-ID.A.2

Chore

This task is divided into four parts.

Â

Role 1

Below are dot plots for three different data sets. The standard deviations for these 3 data sets are given in the post-obit table. Looking at the dot plots and without calculating the standard deviations, match the data sets to the standard deviations.

                      Â

Standard deviation	Information Prepare
5.9	Â
3.3	Â
two.iii	Â

Part 2

Â

1. Draw four rectangles for which the standard divergence of the 4 rectangle heights would be equal to 0.

2. Draw 4 rectangles for which the standard deviation of the iv rectangle heights would be greater than the standard divergence of the rectangle widths.

   

Â

Role 3

Which of the two histograms beneath represents the data distribution with the greater standard deviation? Explain your pick.

Â

Â

Â

Office 4

1. Write two sets of five different numbers that have the same mean but unlike standard deviations.
   Â

2. Write ii sets of v different numbers that have the same standard deviations but unlike means.
   Â

Â

IM Commentary

The purpose of this task is to deepen student understanding of the standard deviation as a measure of variability in a data distribution. The job is conceptual rather than computational and does not require students to calculate the standard difference.

The task is divided into four parts, each of which asks students to think about standard deviation in a different fashion. Parts 3 and 4 may be hard for students, and you lot may want to allow them piece of work in groups to consummate these parts.

If students struggle with Part 2, ask them what it means for the standard divergence to be equal to goose egg. What does this imply well-nigh the rectangle heights?

Part three addresses a common student fault when thinking about the variability and how the standard departure measures variability. Students often choice the incorrect histogram as the 1 for which the standard deviation would be largest because they are thinking about variability in the heights of the bars in the histogram and not agreement that the standard divergence measures variability in the variable that defines the measurement scale in the histogram.

If students struggle with Role four, you might show them a set of five numbers and ask them what would happen to the mean if you were to add 5 to every number. Then inquire if this would change the variability in the data. This will give them a way to think about how to complete part 4.

Â

Solution

Part 1

Standard difference	Data Set
5.nine	3
3.3	two
2.3	ane

Function ii

Answers will vary. For question one, students should depict a set of iv rectangles that all have the same pinnacle. The widths can vary, or they can all be the same. For question two, the heights of the four rectangles drawn should vary more than the widths.

Function iii

The variability would be greater for the data set displayed in Histogram 2. The values in the information set are more than spread out relative to the mean of the information set. The data values in Histogram 1 cluster more tightly around the mean of that distribution.

Part 4

Answers will vary. One possible answer is shown hither.

1. For instance, the information fix five, 10, fifteen, twenty, 25 has a mean of fifteen. The data prepare 13, fourteen, xv, sixteen, 17 besides has a hateful of 15, but the standard difference is smaller.

ii. For example, the following two data sets will accept the same standard deviations merely are centered in different places. Even though they are centered in unlike places, the spread effectually the center is the same.

                        two, 4, 6, eight, 10

                        15, 17, xix, 21, 23

Â

Agreement the Standard Difference

This task is divided into 4 parts.

Â

Part 1

Beneath are dot plots for three unlike data sets. The standard deviations for these three data sets are given in the following table. Looking at the dot plots and without computing the standard deviations, lucifer the information sets to the standard deviations.

                      Â

Standard departure	Information Set
5.9	Â
3.3	Â
2.3	Â

Part two

Â

1. Describe four rectangles for which the standard departure of the 4 rectangle heights would be equal to 0.

2. Draw 4 rectangles for which the standard deviation of the 4 rectangle heights would exist greater than the standard deviation of the rectangle widths.

   

Â

Part 3

Which of the two histograms below represents the information distribution with the greater standard difference? Explain your choice.

Â

Â

Â

Part 4

1. Write two sets of v different numbers that take the aforementioned mean but unlike standard deviations.
   Â

2. Write 2 sets of 5 different numbers that have the same standard deviations but different means.
   Â

Â

cooksuchied.blogspot.com

Source: https://tasks.illustrativemathematics.org/content-standards/tasks/1886

Share this post