1. Common Core State Standards 9 – 12 Mathematics
Quantitative Data
Common Core State Standards 9 – 12 Mathematics

2. Two Main Uses of Statistics
TO DESCRIBE (Data Analysis) TO PREDICT (Statistical Inference)
In secondary math, we focus mainly on descriptive stats, but inference is introduced very informally in the 7th grade where students look at the overlap of two distributions to see if they appear to be different. This, along with the study of the normal distribution and of probability, lays the groundwork for formal statistical inference that is studied in AP Statistics.

3. Describe Data Graphically
Quantitative Data
Boxplot
One-Variable:
Dotplot
Histogram
Quantitative data is data measured or identified on a numerical continuum. The Common Core State Standards deal with both one- and two-variable data. Numerical data can be represented by a variety of graphs, but in the high school Common Core standards we deal with only three types for one-variable data: dotplots, boxplots, and histograms.

4. Describe Data Graphically
Quantitative Data
Two-Variable:
Scatterplot
For two-variable data, we construct and analyze scatterplots.

5. Describe Data Numerically
Quantitative Data
Measures of Center
Measures of Variability
In addition to using graphs to describe data, we also use numerical measures. For categorical data, the measures include counts of how many data values fall into each category, which can be converted to proportions and percents. For quantitative data, we have a plethora of numerical measures – so many that we organize them by type. In the Common Core standards we discuss measures of center and measures of variability.

6. What is the typical value?
Measures of Center
What is the typical value?
One-Variable:
Mean
Median
Mode
Measures of center answer the question: What is the typical value in a data set? If I had to choose one number to describe the data set, what would it be? For one-variable data, the measures of center are the mean, median, and mode. The mean is the arithmetic average, the median is the middle number when the data is ordered, and the mode are the number or number(s) that occur most often. We do not teach the mode since it is rarely used.

7. What is the typical value?
Measures of Center
What is the typical value?
Two-Variable:
Least-Squares Regression Line (LSRL)
For two-variable data, we do not have just one number to describe the center of the data. However, the LSRL is the line that goes through the “center” of the data. So for any given value of x, we can describe and predict the typical y-value that corresponds to it.

8. Measures of Spread How much do values typically vary from the center?
One-Variable: Range Interquartile Range (IQR) Mean Absolute Deviation (MAD) Standard Deviation
Measures of spread answer the question, how much do the values in a data set typically vary from the measure of center? How far off can I expect most data values to be? For one-variable data, the measures of spread are the range, interquartile range, mean absolute deviation, and standard deviation.

9. Measures of Spread How much do values typically vary from the center?
Two-Variable: Correlation Coefficient
For two-variable data, we measure the spread of the points around the LSRL by the correlation coefficient.

10. KWL Chart What do you know about describing one-variable data?
What do you want to know about describing one-variable data?
What have you learned about describing one-variable data?
Now that you have the big picture of statistics, we will focus our tasks on descriptive statistics for one-variable data. Before we begin, we would like for you to take a moment to complete a KWL chart. Answer the questions what do you know about describing one-variable data, and what do you want to know about describing one-variable data?
Have each group share out something they want to know. Write down notes on their responses. Address these at the end of the session.

11. Describing Distributions
Shape
Center
Spread
Outliers
In order to describe a distribution, we address the following things: shape, center, spread, and outliers. This description extends what is done in middle school where they discuss peaks, gaps, and clusters. First let’s talk about the possible shapes of a distribution.

12. Describe Shape Skewed Left Mound-shaped and symmetrical (Normal)
Rectangular (Uniform)
Skewed Right
The four types that are most common are mound-shaped and symmetrical, skewed left or right, and rectangular. Why do you think the shapes were given these names?
Discuss why each shape is given the name that it has. Why is it “normal”? Why “uniform”?

13. Describing Distributions
Shape
Center
Spread
Outliers
What shape does this distribution have? When describing a distribution, we also address the center or most typical value. This can just be “eye-balled” from the graph. What is the approximate center for this graph? Later we will talk about calculating measures of center. Next we describe how spread out the data is – we will just use the range for now and get into calculating measures of spread later. What is the range for this set of data? Finally, if there are outliers - values that seem to fall outside the main cluster of the distribution, we note them. Do you see any outliers here?
So when we describe this set of data from its graph, we could say that the distribution of scores is approximately normal with a center of 45. The scores vary from 0 to There are no apparent outliers.

14. Describing Data
Write a description of the distribution. Be sure to address shape, center, spread, and outliers.
Have a few groups share their description.

15. Do Younger People Have Better Balance?

16. Balancing on One Foot Follow this procedure:
Comfortably balance on your dominant foot before closing your eyes.
Start timing when you close your eyes and say “go”.
Record your time in seconds. Collect your group’s data on the index card and bring to the front – be sure to list which group each value belongs to (over or under 30).
Group Roles: timer, recorder, balancer (rotate roles until each person has had a chance to balance on one foot)
Divide into groups of over and under 30 to create the two data sets.
Time ends when you:
Put your other foot down
Open your eyes
Touch an object for balance with hand or foot
Balance for more than 2 minutes
Person can wiggle, but not hop or spin.

17. Do Younger People Have Better Balance?
Human Boxplot
Note: You will need to set up a number line from 0 to 60, counting by twos or fives out in the hallway. Make sure that you spread out the tick marks far enough to accommodate a person.
Ask teachers to go into the hall and stand on the number line corresponding to the number of seconds you were able to stand on one foot. If two or more teachers have the same number of years, they should stand behind each other, making a “stack” like in a dotplot. Point out to them that they have made a human dotplot.
Give the max/min signs to the teachers on the ends. Next, ask them to find the exact middle number of seconds. If this corresponds to one person (odd number in data set) give this person the median sign. If the exact middle falls between two people (even number in data set), ask the two people in the middle to both hold the median sign. Regardless of whether there is an even or odd number of people, take one out and ask the group what the new median would be. Put that person back in the human plot.
So now we have divided our group into two equal groups. That was so much fun, lets do it again!
Next, tell the group that we are going to divide again. Now we want to know where the middle of the top half is and where the middle of the bottom half is. Let them find these values and give them the Q1 and Q3 signs. Tell them that the “Q” stands for quartile.
Why do you think we call these quartiles? Discuss how the quartiles divide the data set into four equal groups. Discuss terminolgy – lower quartile, middle quartile, upper quartile.
We have the same number of people in each group, but are they spread out the same amount? Emphasize that a boxplot gives a very clear representation of the spread of the data, by showing how far each fourth is spread out.

18. Boxplots Min Q1 Median Q3 Max Lower Upper
Quartile Quartile
The median is the midpoint of an ordered list of data. Half the values are below the median and half are above the median. It is represented by the line in the box.
Discuss how the boxplot is drawn using the five-number summary (min, Q1, M, Q3, max): The numbers that we were looking at are the values of what is called the five-number summary. The five-number summary consists of the min, Q1, M, Q3, and the max. These numbers give a “summary” of the data and when graphed give a picture of the shape, center, and spread of the data set. In a boxplot, all five numbers are represented by vertical lines – the ones for the middle three are drawn longer and connected to make the “box.” Then lines connect each quartile to the min/max. Does anyone remember the longer name for a boxplot? Why was it called box and whisker plot?

19. Boxplots on the Calculator
Please enter the number of seconds each person below 30 years old stood on one foot in List 1 of your calculator and the data of the people above 30 in List 2. Refer to the handout for directions if needed.
Use the number of years of teaching experience data to construct a boxplot on the calculator. Point out that the directions are included in their packet. Note that there are two types of boxplots to choose from on the calculator – one shows outliers, the other doesn’t.
The first choice  , the modified boxplot, indicates outliers by using a dot. The whisker is drawn from a quartile to the most extreme point that is not an outlier.

20. Histograms on the Calculator
Now construct a histogram from the data. Discuss how to select an appropriate window for the data. Use the max and min to help determine the values to set the Xmin and Xmax. Based on these, choose an appropriate interval length (count by 5s , 10s, 0.1 etc.) A histogram should have between 5 and 10 bars. Less than five hides the features of the distribution, more than 10 is too busy for the eye.
Discuss how the shapes of the boxplot and histogram are related. How can you tell whether a distribution is skewed or not by looking at the boxplot?

21. Describing Data
Write a description of the data on the number of seconds each group can stand on one foot. Be sure to address shape, center, spread, and outliers in context.
Have a few groups share their description.

22. How do you know if a data point is an OUTLIER?
So how does the calculator decide whether a data value is an outlier or not?

23. Boxplots Min Q1 Median Q3 Max Lower Upper
Quartile Quartile
The Interquartile Range (IQR) is the spread of the middle 50% of the data. It is represented by the length of the box.
In order to talk about outliers, we need to first talk about the variation of the data values near the center, which is called the Interquartile Range. Given the word “interquartile” how do you think the IQR is calculated?
The IQR is the difference between the upper and lower quartiles.
As a measure of spread, the IQR tells us how far from the mean we can expect values to be. So we use this as a basis to determine how far out a value needs to be in order to be called an outlier. What is the IQR for our data?

24. Statisticians devised what is called the 1
Statisticians devised what is called the 1.5IQR rule to identify outliers.
The first step is to calculate the IQR, which we just did.
Then we multiply the IQR by What is this amount?
Add this number to Q3. Any value above this amount is considered an outlier. Do we have any high outliers?
Then subtract that number from Q1. Any value below this amount is an outlier. Do we have any low outliers?
Why 1.5? John Tukey, the statistician who devised this rule, is quoted as saying that “one was not enough and two was too many. “

25. Matching Tasks
In groups, have participants match the histograms to the boxplots that were made from the same set of data. The data sets are the homework scores of five students.

26. Measures of Center
So, we have finished talking about the graphical representation of one-variable data. Now we will go into more depth on the numerical descriptions of one-variable data, starting with measures of center.

27. What is the typical value?
Measures of Center
What is the typical value?
One-Variable:
Mean
Median
Mode
Remember, measures of center describe the typical value for a data set. We just found the median for our set of data. Generally speaking how would you describe the median?
In general, the median is the middle number when a data set is put in order. The median can be part of the data set, but does not have to be. It is the number that divides the data set so that 50% of the data is above the median and 50% is below the median.

28. Mound-shaped and symmetrical (Normal)
Mean vs. Median
Mound-shaped and symmetrical (Normal)
Skewed Left
Skewed Right
Ask them to discuss with a shoulder buddy where the mean and median for each distribution would be located. They should make a quick sketch on their paper and mark where these values would be.
What is the location of the mean relative to the median in each type of distribution? Why does this happen?
In a symmetrical distribution, the mean and the median are approximately equal. In skewed distributions, the outliers tend to “pull” the mean towards them, in order to maintain the mean as a balance point for the data set.

29. What about Variability?
Why do we need to know how spread out the data is?

30. What about variability?
Consider the following test scores: Who is the best student? How do you know?
Student
Test 1
Test 2
Test 3
Test 4
Johnny 65 82 93
100
Will 81 86 88 83
Anna 80 99 72
Let teachers discuss this in their groups and make a decision. Have them share out their ideas.

31. What about variability?
Student
Test 1
Test 2
Test 3
Test 4
Test
Average
Johnny 65 82 93
100 85
Will 81 86 88 83
Anna 96 77
Usually we calculate the average to describe how a student is doing. However, these 3 students all have the same average. But we just agreed that they are not “equal” in their test performance. We need more information than just the typical test score, we need to know how consistent each student is – measures of spread will give us that information. (There is also the issue of performance over time, but we will not be addressing that here.)

32. Measures of Spread How much do values typically vary from the center?
One-Variable: Range Interquartile Range (IQR) Mean Absolute Deviation (MAD) Standard Deviation
Remember, measures of spread address how far most of the data values typically fall from the center. We have already discussed the range and the interquartile range. Lets talk about the mean absolute deviation and the standard deviation. What does the word “deviation” mean?

33. So what exactly is deviation?-4 -3 +5 -1 +3
Remember that the mean is the balance point for a set of data:
For example, the mean of 1, 2, 4,8, and 10 is 5 because it balances the distances on a number line from the mean to each data value.
(-4) + (-3) + (-1) = -8
(+5 ) + (+3) = +8

34. So what exactly is deviation?
(-4) + (-3) + (-1) + (+5) + (+3)
= 0  5 = 0
An average deviation of zero means that there is no variability!
Houston, we have a problem!
In order to have a numerical measure of spread, it would make logical sense to find the average distance, or deviation of each value from the mean. So we would add up the deviations and divide by the number of points.

35. How can we fix our problem?
Take the absolute value of each distance/deviation and then find the average
So the average distance or deviation from the mean is about 3 points (above or below).
This is called the Mean Absolute Deviation, or MAD

36. Mean Absolute Deviation
-3.2
+3.2
Let’s go back to our number line to visualize this. Most of the data falls within plus or minus 3.2 points of 5.

37. This is called the standard deviation.
Square each deviation and then find the average of the squared deviations
This is called the standard deviation.
What else can we do? We could square each distance to make it positive and then find the average of the squared deviations.
But wait, 12 is a lot bigger than the 3.2 we found by calculating the mean average deviation.
Why is this? What happened?
We need to undo our squaring by taking the square root – then we will have a number that makes sense.

38. Back to Johnny, Will and Anna . . .
Without using the formula, calculate the standard deviation for each student.
Student
Test 1
Test 2
Test 3
Test 4
Johnny 65 82 93
100
Will 81 86 88 83
Anna 80 99 72
They should use the sheet in the packet to guide them.

39. Who is the best student? How do you know?
Test 1
Test 2
Test 3
Test 4
Test
Average
Standard Deviation
Johnny 65 82 93
100 85
13.2
Will 81 86 88 83
2.7
Anna 80 99 72
10.0
Let teachers discuss this in their groups and make a decision. Have them share out their ideas.

40. Standard Deviation
Here is the formula for standard deviation. Hopefully now it doesn’t look so scary!
Show how to find all measures using 1-Var stats on the calculator.

41. How do you decide whether to use the mean and standard deviation or the median and IQR to summarize the data numerically? Outliers
In general, which one is less sensitive to outliers, the median or the mean? Why? (mean and standard deviation are more sensitive because their formulas take every data value into account; the median and IQR do not)
If it comes up/if there is time:
Whether to leave an outlier in the analysis depends on close inspection of the reason it occurred.
-If it was the result of an error in data collection or entry it should be corrected if possible, and if not, removed.
-If it is fundamentally unlike the other values, it should be removed from the data set.
-If it is simply an unusually large or small value, you have two choices:
-Report measures of center and spread that are resistant to outliers.
-Do the analysis twice, with and without the outlier, and report both.

42. Comparing Two Sets of Data
Do younger people have better balance?
How can we answer this question? What experiment can we do? What would we measure?

43. Creating Histograms
Using technology, create a histogram of each set of data. Make sure you use the same scale for each!

44. Side-by-Side Boxplots
Using technology, create side-by-side boxplots.

45. Comparing The Data Sets
Write a description comparing the two data sets in context. Address shape, center, spread, and outliers. Answer the question, “Who balances better?”
Which representation gives you more information?
Which numerical measures best describe the data sets? Explain.
Share out and discuss.

46. Mathematical Practices
Considering practices 1, 3, and 6 What behaviors from these practices have you exemplified while performing these tasks?

47. KWL Chart What do you know about describing Statistics?
What do you want to know about describing Statistics?
What have you learned about describing Statistics?
Finish the chart. Go back to earlier list of “want to knows” and see if they were covered in this session.

48. “What task can I give that will build student understanding?”
rather than
“How can I explain clearly so they will understand?”
Grayson Wheatley, NCTM, 2002

49. North Carolina Common Course One

50. Questions and Comments
9/17/2018 • page 50

51. Thank You!!
Contact Information
Secondary Mathematics Consultants
Robin Barbour Johannah Maynor
9/17/2018 • page 51