Question 50·Medium·One-Variable Data Distributions; Measures of Center and Spread
Each dot plot shows the number of minutes of piano practice completed by each student in one week for two different clubs.
Which statement best compares the standard deviations of the numbers of minutes of piano practice completed by each student for these two clubs?
When comparing standard deviations from graphs, first check whether one data set is a simple transformation of the other. Adding the same constant to every value shifts the data but keeps the standard deviation the same, while multiplying every value by a constant stretches or shrinks the data and scales the standard deviation by that constant. Use that transformation idea instead of trying to calculate standard deviation directly.
Hints
Look for a pattern between the plots
Check whether the Club Q values are obtained from the Club P values by adding the same number or multiplying by the same number.
Focus on how spread changes under transformations
Think about what happens to all distances from the mean if every data value is multiplied by 2.
Connect distances to standard deviation
Standard deviation is a measure of typical distance from the mean, so if those distances scale, the standard deviation scales the same way.
Desmos Guide
Enter the Club P data as a list
Create a list for Club P by repeating values according to the dots.
For example, enter
P={9,10,10,11,11,11,12,12,12,13,13,14}
Create the Club Q list by scaling
Enter Q=2P to double every value in the list.
Compare standard deviations in Desmos
Compute stdev(P) and stdev(Q).
Observe that stdev(Q) is larger (in fact, it is twice stdev(P)), so Club Q has the greater standard deviation.
Step-by-step Explanation
Relate the two dot plots
Read corresponding tick marks: the Club P values are , while the Club Q values are .
Each Club Q value is times a Club P value (for example, and ), and the dot counts at the corresponding positions match.
Use the scaling rule for spread
Multiplying every value in a data set by multiplies every distance from the mean by .
Since standard deviation measures a typical distance from the mean, the standard deviation is also multiplied by .
Compare the standard deviations
Because Club Q's data set is a scaling of Club P's data set, Club Q has the greater standard deviation.
Answer: The standard deviation of the number of minutes of piano practice completed by each student for Club P is less than the standard deviation of the number of minutes of piano practice completed by each student for Club Q.