Dear Parents and Caregivers,

Thank you for supporting your child to achieve success in school. We value your input and active participation in your child’s education. These letters are designed to help you understand the work your child brings home and the academic expectations of Arizona’s College and Career Ready Standards. Your child is developing the necessary skills and knowledge to help them compute, think, and reason mathematically. This letter is about data distribution in sixth grade. End-of-year goals In sixth grade, students begin to develop their ability to think statistically. They find and interpret the measures of center (mean, median, mode, range) and recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for describing and summarizing data sets. Vocabulary • Absolute value: the distance of a number from zero on the number line; absolute value is always positive. • Mean: the value that each data point would take on if the total of the data values was redistributed equally. The mean can be found by adding together all the numbers in the data set, and dividing by the number of values (sometimes referred to as the average). • Median: the middle value of a data set • Mode: the value of the data that occurs most frequently (Some data sets may not have a mode, or have more than one mode.) • Range: the difference between the greatest and the least value in a data set • Quartile: the values that divide a set of data into quarters • Interquartile range: the difference between the first quartile and the third quartile of a set of data; this is one way to describe the spread of the data. • Mean absolute deviation: the average distance between each data value and the mean • Variability: the extent to which data points in a data set vary from the mean (average) value or from each other Understanding the Measures of Center Students collect real-world data and then describe the data using different measures. The mean or average is one commonly used statistic, which may be familiar. Students will learn that the mean is a measure that describes the value if the data in a set were “leveled out”. For example, this data set was generated when eight students counted the number of letters in their first names. (4, 5, 5, 6, 4, 4, 5, 7) Stacking cubes, with each stack representing one student’s name, could represent this data set. The data is ordered from least to greatest.

Students can model the mean by “leveling” the stacks or distributing the blocks so the stacks are even. One block from the stack of 6 and two blocks from the stack of 7 can be moved down to the stacks of 4, so that all the stacks have five blocks. Students are seeking to answer the question, “If all of the students had the same number of letters in their names, how may letters would each person have?” (five)

Mesa Public Schools/Grade 6/datadistribution/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-2014).

If it is not possible to make the stacks exactly even, students could consider what fraction of the extra blocks each stack would have. Students will make the connection between this model and the calculation for finding mean. The mean that can be found by adding together all the numbers in the data set, and dividing by the number of values in the set.

Using this same data set, the student could identify the median value of the set. By placing the data in order from least to greatest, the median would be identified as the middle number. Since this data set has eight values, the median would be between the two numbers in the center of the data set (4, 4, 4, 5, 5, 5, 6, 7).

Median Those numbers (5, 5) would be added together and divided by 2 to find the middle number (5 + 5 = 10; 10 ÷ 2 = 5). The median for this set is 5. To find the mode of this set, the student would look for the numbers that occur most often. In this set 4 occurs three times, 5 occurs 3 times, 6 occurs once and 7 occurs once. Since 4 and 5 both occur three times this set has two modes, 4 and 5. To find the range of the set of data, a student would find the difference between the greatest value and the least value in the set. (7 – 3 = 4; 4 is the range.) Understanding Measures of Variability Students may use real-world data to explore quartiles and mean absolute deviation as ways to describe the data. The data set below shows the number of cans of food collected by seven classrooms for the food drive. The lower quartile (LQ) and upper quartile (UQ) are labeled. 14, 18, 19, 20, 24, 29, 31

Lower quartile Median Upper quartile This shows that one quarter of the data lie below the lower quartile and one quarter of the data lie above the upper quartile. The upper and the lower quartiles are the medians of the upper half and the lower half of the data, respectively. The interquartile range is the difference between the upper quartile and the lower quartile (29 – 18 = 11; interquartile range is 11). Students may also use the mean absolute deviation to describe the variability in a data set. It describes the average distance between each data value and the mean. For example, this data set was generated by counting the number of contacts eight students had in their cell phones. (52, 48, 60, 55, 59, 54, 58, 62) To find the mean absolute deviation, first find the mean of the set (56). Then, find the distance between each data value and the mean by finding the absolute value of the difference between each data value and the mean. (Subtract each value from the mean and write the absolute value of each difference.) The mean number of contacts stored and the distance each data value is from the mean is shown below. An x represents each data value.

Finally, find the average of the distances. So, the average distance between each data value and the mean is 3.75 contacts.

How to help at home • Watch these videos about data distribution from LearnZillion. http://learnzillion.com/lessonsets/132-understand-and-describe-the-distribution-of-a-set-of-data • Remember, making mistakes is a part of learning.

Mesa Public Schools/Grade 6/datadistribution/2013 Authorization to reprint or disseminate must be granted by Mesa Public Schools (February-2014).