Understanding Statistical Questions and Variability in Data

1. Statistics Unit 6

2. MCC6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. Examples: Statistical questions that anticipate variability Statistical questions that do not anticipate variability Vocabulary Words Statistical Question Variability

3. What is a statistical question? A question that generates a variety of answers is called a statistical question. Depending on the question, the type of data gathered can be either categorical or numerical. An example of a categorical question is “What is your favorite type of pizza?” The answers generated by this question will be categories of pizza types such as pepperoni, cheese, or sausage. An example of a numerical question is “How many pencils does each member of our class have in his or her desk?” A variety of numerical answers about the number of pencils would be given by a typical 6th grade class.

4. Statistical Variability Identify if the statement below is a statistical question? What is the height of each person in my class? What are the math test scores of the students in my class? How old am I? How many letters are in my name? How many letters are in the names of each person in my class? How many pets are owned by each student in my grade level? What is my math test score? What is my height? Examples of Statistical Questions: Non-Examples of Statistical Questions:

5. MCC6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. Examples: Describe a set of data by its center Describe a set of data by its spread Describe a set of data by its overall shape Vocabulary Words Center: mean, median, mode Spread: range, mean absolute deviation Shape: cluster, gap, outlier

8. Shape of Data

9. Center of Data Mean, Median, Mode Fifteen students were asked to rate how much they like Middle school on a scale from one to ten. Here is the data collected: 1, 10, 9, 6, 5, 10, 9, 8, 3, 3, 8, 9, 7, 4, 5 The first step is to put your data in ascending order. 1, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10 Let’s find the Mean = average value for the data 1 + 3 + 3 + 4 + 5 + 5 + 6 + 7 + 8 + 8 + 9 + 9 + 9 + 10 + 10 15 97 15 = = 6.5

10. Center of Data Mean, Median, Mode Fifteen students were asked to rate how much they like Middle school on a scale from one to ten. 1, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10 Let’s find the Median = the value for which half the numbers are longer and half the numbers are smaller. Let’s find the Mode = the number that occurs most often Mean: 6.5 Median: 7 Mode: 9

11. Variability of Data Mean Absolute Deviation Fifteen students were asked to rate how much they like Middle school on a scale from one to ten. 1, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10 Range Fifteen students were asked to rate how much they like Middle school on a scale from one to ten. 1, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10 MEAN | 6.5 – 1 | + | 6.5 – 3 | + | 6.5 – 3 | + | 6.5 – 4 | + | 6.5 – 5 | + | 6.5 – 5 | + | 6.5 – 6 | + | 6.5 – 7 | + | 6.5 – 8 | + | 6.5 – 8 | + | 6.5 – 9 | + | 6.5 – 9 | + | 6.5 – 9 | + | 6.5 – 10 | + |6.5 – 10| = 36.5 Let’s find the Range = Difference between maximum and minimum data. 10 – 1 = 9 Range = 9 36.5 15 = 2.43

12. MCC6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Examples: How a number that describes the measure of center is different from a number that describes the measure of variation Vocabulary Words Center Variation

13. Center Vs Variability Center: Summarizes all the values with a single number • Median: the middle number • Mean: the average number Variability: Describes how all the values vary with a single number • Range: shows the greatest amount of variation between two data values • MAD: shows the average variation between the data values

14. MCC6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Examples: Draw a dot plot Draw a histogram Draw a box plot Vocabulary Words Dot plot Histogram Box Plot

16. Box Plot

18. Box Plot How to Create a Box-and-Whisker Plot 1) Write the data in order from least to greatest. 2) Draw a horizontal number line that can show the data in equal intervals. 3) Find the median of the data set and mark it on the number line. 4) Find the median of the upper half of the data. This is called the upper quartile (Q3). Mark it on the number line. 5) Find the median of the lower half of the data. This is called the lower quartile (Q1). Mark it on the number line. 6) Mark the lower extreme (minimum) on the number line. 7) Mark the upper extreme (maximum) on the number line. 8) Draw a box between the lower quartiles and the upper quartile. Draw a vertical line through the median to split the box. 9) Draw a “whisker” from the lower quartile to the lower extreme. 10) Draw a “whisker” from the upper quartile to the upper extreme.

19. Dot Plots

20. Dot Plots How to Create a Dot Plot 1) Draw a horizontal number line. 2) Determine and mark a scale of numbers below the line. Make sure to include the minimum and maximum values in the data set and all consecutive number values in between. Example: In the data set, there is a minimum value of 1 and a maximum value of 5. The number line must include tick marks for every number value from 1 through 5. A few numbers before the minimum and a few numbers after the maximum can be included. 3) A dot is tallied for each value above the corresponding number. Keep the imaginary y-axis as a frequency mark to ensure that dots are plotted correctly. 4) Put a title on the graph.

21. Histogram

22. Histogram How to Create a Frequecty Chart & Histogram 1) Make a frequency table of the data by selecting a range that will contain all of the data and then divide it into equal intervals. In the example above, the range of ages is from 0 to 69 so equal intervals of 10 years were selected. 2) Using graph paper, draw an x-axis where each box will represent an interval of numbers to represent the ranges. 3) Draw a y-axis with a scale of numbers appropriate for the data. Common scales are multiples of 1, 2, 5, 10 or 20. 4) Draw each bar on the histogram to correlate the intervals with the frequency of occurrence. 5) Title the graph and the x and y-axis.

23. MCC6.SP.5 Summarize numerical data sets in relation to their context, such as by: Examples: Draw a frequency chart to report the number of observations Describe a graph or information on how it was measured and its unit of measurement MCC6.SP.5a Reporting the number of observations. MCC6.SP.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. Vocabulary Words Frequency Chart

24. Reporting # of Observations How many families have 3 children? 12 families How many families have less than 3 children? 11 families

25. Describing the Units What information was collected to create this graph? • 28 families with 1 – 5 children were surveyed • Families reported how many children they had What are the units of measurement used within this graph? • Number of children in families • Number of families with specified amount of children

26. MCC6.SP.5 Summarize numerical data sets in relation to their context, such as by: Examples: Describe a box plot in terms of median and interquartile range Describe a histogram in terms of median, mean, and mean absolute deviation Describe a dot plot in terms of mean, median, and mean absolute deviation MCC6.SP.5c 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 was gathered. Vocabulary Words Outlier

27. Measures of Center

28. Measures of Variability

29. MCC6.SP.5 Summarize numerical data sets in relation to their context, such as by: Examples: How does the measure of center relate to the shape of the data set? How does the measure of variability relate to the shape of the data set? MCC6.SP.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered.

30. Representative Measures of Center Use the Dot Plot to identify the Mean, Median & Mode Mode = Number used most in the set 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5 Mode = 1 Median = Number in middle of set 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5 Median = 2 Mean = Average 1+1+1+1+1+1+1+1+2+2+2+2+2+3+3+3+4+4+4+5= 44 44/20 = 2.2 Mean = 2.2

31. Representative Measures of Variability Use the Dot Plot to identify the Range, IQR, and MAD IQR: 1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,5 1st Quartile: 1 3rd Quartile: 3 IQR: 3 – 1 = 2 Range = 5 – 1 = 4

32. Math Resource