# 6.15 and *6.11 - Measure of Center

6.15    The student will

a)  describe mean as balance point; and

b)  decide which measure of center is appropriate for a given purpose.

6.11  The student will

a)  represent the mean of a data set graphically as the balance point; and

b)  determine the effect on measures of center when a single value of a data set is added, removed, or changed.

Bloom's Level:  Understand, Analyze

### BIG IDEAS

• I can determine what grade I must make on the next test in order to raise my grade to be eligible to play sports.
• I will be able to interpret measures in the context of a given situation and identify what is normal for any set of data.

### UNDERSTANDING THE STANDARD

• Measures of center are types of averages for a data set. They represent numbers that describe a data set. Mean, median, and mode are measures of center that are useful for describing the average for different situations.
• Mean works well for sets of data with no very high or low numbers.
• Median is a good choice when data sets have a couple of values much higher or lower than most of the others.
• Mode is a good descriptor to use when the set of data has some identical values or when data are not conducive to computation of other measures of central tendency, as when working with data in a yes or no survey.
• The mean is the numerical average of the data set and is found by adding the numbers in the data set together and dividing the sum by the number of data pieces in the set.
• In grade 5 mathematics, mean is defined as fair- share.
• Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean. This is the concept of mean as the balance point.
• Defining mean as balance point is a prerequisite for understanding standard deviation.
• The median is the middle value of a data set in ranked order. If there are an odd number of pieces of data, the median is the middle value in ranked order. If there is an even number of pieces of data, the median is the numerical average of the two middle values.
• The mode is the piece of data that occurs most frequently. If no value occurs more often than any other, there is no mode. If there is more than one value that occurs most often, all these most-frequently-occurring values are modes. When there are exactly two modes, the data set is bimodal.

·  Categorical data can be sorted into groups or categories while numerical data are values or observations that can be measured. For example, types of fish caught would be categorical data while weights of fish caught would be numerical data.

·  Measures of center are types of averages for a data set. They represent numbers that describe a data set. Mean, median, and mode are measures of center that are useful for describing the average for different situations.

–  Mean may be appropriate for sets of data where there are no values much higher or lower than those in the rest of the data set.

–  Median is a good choice when data sets have a couple of values much higher or lower than most of the others.

–  Mode is a good descriptor to use when the set of data has some identical values, when data is non-numeric (categorical) or when data reflects the most popular item.

·  Mean can be defined as the point on a number line where the data distribution is balanced. This requires that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances from the mean of all the data points below the mean. This is the concept of mean as the balance point.

-  Example: Given the data set:

2, 3, 4, 7

The mean value of 4 can be represented on a number  line as the balance point:

·  The mean can also be found by calculating the numerical average of the data set.

·  In grade five mathematics, mean is defined as fair share.

·  Defining mean as the balance point is a prerequisite for understanding standard deviation, which is addressed in high school level mathematics.

·  The median is the middle value of a data set in ranked order. If there are an odd number of pieces of data, the median is the middle value in ranked order. If there is an even number of pieces of data, the median is the numerical average of the two middle values.

·  The mode is the piece of data that occurs most frequently. If no value occurs more often than any other, there is no mode. If there is more than one value that occurs most often, all these most-frequently-occurring values are modes. When there are exactly two modes, the data set is bimodal.

### ESSENTIALS

• What does the phrase “measure of center” mean?
This is a collective term for the 3 types of averages for a set of data – mean, median, and mode.
• What is meant by mean as balance point?  Mean can be defined as the point on a number line where the data distribution is balanced. This means that the sum of the distances from the mean of all the points above the mean is equal to the sum of the distances of all the data points below the mean. This is the concept of mean as the balance point.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.15a1  Find the mean to a set of data.

6.15a2  Identify and draw a number line that demonstrates the concept of mean as a balance point for a set of data.

·  6.11a1  Represent the mean of a set of data graphically as the balance point represented in a line plot.

6.15b1  Describe the 3 measures of center and a situation in which each would best represent a set of data.

·  6.11b1  Determine the effect on measures of center when a single value of a data set is added, removed, or changed.

### KEY VOCABULARY

mean, balance point, measure of center, measure of central tendency, median, mode, survey, fair share, sum, standard deviation, outlier, ranked order

Updated: Oct 27, 2017