Calculating Measures of Center and Variability
5 min
Narrative
The purpose of this Warm-up is to encourage students to review how to calculate mean and median.
Launch
Display one problem at a time. Give students 1 minute of quiet think time followed by a whole-class discussion.
Student Task
Decide if each situation is true or false. Explain your reasoning.
- The mean can be found by adding all the numbers in a data set and dividing by the number of numbers in the data set.
-
The mean of the data in the dot plot is 4.
- The median of the data set is 9 for the data: 4, 5, 9, 1, 10.
-
The median of the data in the dot plot is 3.5.
Sample Response
- True. Sample reasoning: The statement is the correct way to calculate the mean.
- False. Sample reasoning: The mean is 3.5 because the sum is 28 and 28÷8=3.5. Do not round to 4.
- False. Sample reasoning: Nine is not the median because the numbers are not arranged in order.
- True. Sample reasoning: The median is 3.5 because the two values in the middle are 3 and 4, and their mean is 3.5.
Activity Synthesis (Teacher Notes)
The goal of this activity is to review how to calculate mean and median and to identify common mistakes in the calculation of mean and median. In the discussion, ask students to recall what information the mean and median reveal about the data.
- “What does the mean tell you about the data?” (On average, where the center of the data is.)
- “What does the median tell you about the data?” (Half of the numbers are greater than or equal to the median, and half are less than or equal to the median.)
Some common mistakes to avoid:
- Not putting the numbers in order when finding the median.
- If two numbers are in the middle, not adding them and dividing by two to find the median.
- Finding the middle number on the horizontal axis rather than in the data.
- Rounding the mean or median (if it must be calculated) to the nearest whole number.
Anticipated Misconceptions
Some students may forget to sort the data when finding the median. Ask them, “What is a median? What does it tell you about the data?” Some students may not remember how to find the median when there is an even number of data values. Ask them, “What does the median tell you about the data? How could we find a middle number between these two values?”
Standards
- 6.SP.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- 6.SP.B.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- HSS-ID.A.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- HSS-ID.A.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
10 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Calculating Measures of Center and Variability
5 min
Narrative
The purpose of this Warm-up is to encourage students to review how to calculate mean and median.
Launch
Display one problem at a time. Give students 1 minute of quiet think time followed by a whole-class discussion.
Student Task
Decide if each situation is true or false. Explain your reasoning.
- The mean can be found by adding all the numbers in a data set and dividing by the number of numbers in the data set.
-
The mean of the data in the dot plot is 4.
- The median of the data set is 9 for the data: 4, 5, 9, 1, 10.
-
The median of the data in the dot plot is 3.5.
Sample Response
- True. Sample reasoning: The statement is the correct way to calculate the mean.
- False. Sample reasoning: The mean is 3.5 because the sum is 28 and 28÷8=3.5. Do not round to 4.
- False. Sample reasoning: Nine is not the median because the numbers are not arranged in order.
- True. Sample reasoning: The median is 3.5 because the two values in the middle are 3 and 4, and their mean is 3.5.
Activity Synthesis (Teacher Notes)
The goal of this activity is to review how to calculate mean and median and to identify common mistakes in the calculation of mean and median. In the discussion, ask students to recall what information the mean and median reveal about the data.
- “What does the mean tell you about the data?” (On average, where the center of the data is.)
- “What does the median tell you about the data?” (Half of the numbers are greater than or equal to the median, and half are less than or equal to the median.)
Some common mistakes to avoid:
- Not putting the numbers in order when finding the median.
- If two numbers are in the middle, not adding them and dividing by two to find the median.
- Finding the middle number on the horizontal axis rather than in the data.
- Rounding the mean or median (if it must be calculated) to the nearest whole number.
Anticipated Misconceptions
Some students may forget to sort the data when finding the median. Ask them, “What is a median? What does it tell you about the data?” Some students may not remember how to find the median when there is an even number of data values. Ask them, “What does the median tell you about the data? How could we find a middle number between these two values?”
Standards
- 6.SP.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- 6.SP.B.5.c·Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- HSS-ID.A.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- HSS-ID.A.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.1·Represent data with plots on the real number line (dot plots, histograms, and box plots).
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
- S-ID.2·Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.