1. Describing a Distribution
Teachers, be sure to look at the notes section of each slide to additional information.
Lesson 8.1.3

2. Using Visual Representation of Data
Often, data is presented in a chart rather than a list of numbers because a chart is much easier to read.

3. Prior… In Chapter 2, you learned how to visually
display data using dot plots and histograms.
“Old” Miley & Sophie
A teacher asked 18 students in her class…
“How many pets do you own?”
Responses:
0, 4, 2, 1, 6, 2, 0, 1, 2, 6, 3, 2, 4, 1, 7, 2, 2, 8
histogram
dot plot
Review how to read a dot plot… 18 students so 18 dots…. Two students have no pets, three students have 1 pet and so forth...
Review how to read the histogram… = 18 students… five students have 0-1 pets, 7 students have 2-3 pets and so forth…

4. Prior… In this chapter, we have been learning
about the mean, median and range.
A teacher asked 18 students in her class…
“How many pets do you own?”
Responses:
0, 4, 2, 1, 6, 2, 0, 1, 2, 6, 3, 2, 4, 1, 7, 2, 2, 8
MEAN: = 53 = pets
MEDIAN: 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 6, 6, 7, 8
RANGE: 8 – 0 = 8
4÷2=2
8 numbers
8 numbers

5. Prior… We have also been learning about outliers. What is an outlier?
Do you notice any outliers in our “Number of Pets” chart?
Give an example of a “number of pets” that would be an outlier.
Which measure of center will an outlier affect the most?
Is that good or bad?
An outlier is a number that is way different than the other numbers in the data set.
There are no outliers present in this set of data.
An example of an outlier would be if someone owned 30, 50 or 100 pets.
Measures of center include mean and median. An outlier affects the mean much more than the median.
An outlier can be bad if you are using the mean to find the center because an outlier throw off the mean.

6. Prior… An outlier messes up the mean. It can throw it off BIG TIME!
Use the Median.
An outlier messes up the mean. It can throw it off BIG TIME!
Peak
Point out the outliers on each image. Explain again that an outlier is a number that is way different than the other numbers in a data set. Be sure to point out the gap and outlier on the histogram.
Gap

7. Prior… Since our “Number of Pets” survey does not contain an outlier…
The MEAN is the best measurement
of the center. 2.9
Point out that there are not major gaps. All of the numbers are pretty close together.

8. What are we going to do today?

9. Describing a Distribution
It is important to understand that a set of data collected to answer a statistical question has a distribution that can be described by:
1. its center (mean, median)
2. its spread (range, interquartile range, mean absolute deviation)
3. its overall shape
Today…
You will utilize your prior knowledge to help you statistically describe data from a visual representation.
interquartile range and MAD will be in another lesson

10. Let’s Get Started…
Section 1: Describing a data distribution by its shape.
Section 2: Describing a data distribution by its center and variation.
Section 3: Finding the mean, median, and range
on a Dot Plot
Section 4: Finding the mean, median, and range on a Histogram

11. Exploratory
“These bar graphs have the same center and spread, but they are completely different!”
What are some differences that you notice? Work with a partner and list as many as you can.

12. Vocabulary
To accurately describe the shape of a distribution, there is a standard group of vocabulary words that can help us.
(1 min) 25-26

13. Symmetry:
When it is graphed, a symmetric distribution can be divided at the center so that each half is a reflection of the other.
Examples:

14. Peak:
A point on the graph that is higher than the points directly to the left and right.
Examples:
One Peak
Two Peaks

15. Left Skewed and Right Skewed:
Distributions with a tail on the right are said to be skewed right or positively skewed.
Distributions of incomes or wages are often positively skewed because of the presence
of some relatively high values that are not offset by correspondingly low values.
Distributions with a tail on the left are said to be skewed left or negatively skewed.
Examples:
Right Skewed (or Positively Skewed)
Left Skewed (or Negatively Skewed)
Look for the tail !

16. Uniform: A graph that is evenly spread out, with no peaks. Examples:

17. Gap and Outliers:
Distributions may also have unusual features. The two most common ones are:
1) Gap: An area of the distribution where there are no entries in the data set.
2) Outlier: An element of the data set that is much higher or much lower than all the other elements.
Examples:
Outlier
Gap
Gap

18. Looking Back
Let’s go back to the two graphs we analyzed earlier. How can we describe them using the vocabulary that we just learned?
Histogram 1: symmetrical, Peaks at 4, not gaps, no outlier.
Histogram 2: symmetrical, Peaks at 2 and 6, no gaps, no outlier.

19. General Rules
When a distribution is perfectly symmetric, the mean will equal the median.
When data is skewed right, the mean will be “pulled” to the right of the median.
When data is skewed left, the mean will be “pulled” to the left of the median.
** use the median
** use the mean
** use the median
** use the median

20. Moving on… Section 1: Describing a data distribution by its shape.
Section 2: Describing a data distribution by its center and variation.
Section 3: Finding the mean, median, and range
on a Dot Plot
Section 4: Finding the mean, median, and range on a Histogram

21. Center vs. Variation Measures of Variation Measures of Center Range
Interquartile Range
Mean Absolute Deviation
a single number that describes
how the numbers are spread out
Measures of Center
Mean
Median
a single number that
summarizes the value of the data
Go over (or ask students) about the two charts.
Discuss shape, center and spread.
Interquartile range and MAD are a later lesson.

22. The MEAN… what does it really mean?
We know it is the average, but let’s refer back to the pet survey to see what it really means.
If we took all 53 pets and split them out equally to the 18 students doing the survey… each student would get 2.9, or about 3 pets each.
split evenly
A teacher asked 18 students in her class…
“How many pets do you own?”
Responses
0, 4, 2, 1, 6, 2, 0, 1, 2, 6, 3, 2, 4, 1, 7, 2, 2, 8
18 students surveyed

23. The MEDIAN… what does it really mean?
We know that it is the middle number after ordered, but what that means is that:
50% of the responses are to the left of the median &
50% of the responses are to the right of the median.
A teacher asked 18 students in her class…
“How many pets do you own?”
Responses
0, 4, 2, 1, 6, 2, 0, 1, 2, 6, 3, 2, 4, 1, 7, 2, 2, 8 2 0%
100%
50%

24. The RANGE… what does it really mean?
Range is not a measure of the center!
The range is the biggest number minus the smallest number.
This number really indicates how spread out or varied the numbers are in a data set.
The larger the range… the more spread out (or varied) the numbers are!
8 – 0 = 8
A teacher asked 18 students in her class…
“How many pets do you own?”
Responses
0, 4, 2, 1, 6, 2, 0, 1, 2, 6, 3, 2, 4, 1, 7, 2, 2, 8

25. Compare these two dot plots. The height dot plot has the larger range
Compare these two dot plots. The height dot plot has the larger range. This means that the answers to the survey are more spread out and varied. The data makes sense if you think about it. Most students are going to have about the same number of books. However, their heights will be pretty varied!
The height survey has a larger range than the textbook survey. Most students will have about the same amount of books. However, the height of students can vary alot. Students can be 4 feet tall or 6 feet tall…
The larger the range, the more spread out and varied!
spread
spread
Range: 8 – 5 = 3
Range: = 17

26. You Try!
Given below is a statistical question and the histogram of data collected to answer the question. 1) How many movies were recorded in this survey? 2) The shape of the distribution can be described as which of the following? a. uniform c. left skewed b. approximately symmetrical d. right skewed 3) Are there any peaks and if so how many? 4) Are there any outliers? 5) Describe the spread of the data. 6) In which interval would you guess the center is located? Why? 7) What would be the best measure of center for this distribution (mean or median)? Why? 8) Most movies had a length of? a. 120 to 150 minutes long b. 80 to 110 minutes long c. 100 to 130 minutes long d. 110 to 120 minutes long
A local theater was asked… “What was the length of the movies shown over the last six months?”
= 28
Approximately symmetrical
Yes; 2 No
The shortest movie was 80 minutes. The longest movie was 150 minutes. Range: 150 – 80 = 70
Since the shape is approximately symmetrical, the center is pretty easy to figure out. It is probably somewhere in the interval
Mean because the shape is symmetrical and there is not an outlier.
100 to 130 minutes

27. Let’s Get Started…
Section 1: Describing a data distribution by its shape.
Section 2: Describing a data distribution by its center and variation.
Section 3: Finding the mean, median, and range
on a Dot Plot
Section 4: Finding the mean, median, and range on a Histogram

28. MEAN
What does this mean?
If you split out all 30 of the siblings evenly between the 15 people doing the survey…
They would each get 2 siblings.
Show students the normal way . Then show them the short cut as described on this screen.
= = 2

29. MEDIAN Median How do you determine the median in a dot plot?
Again, this is a perfectly symmetrical dot plot, so it is pretty obvious that the middle number is 2. However… let’s go through the steps of justifying it.
Notice that in this dot plot that the data is already ordered. We don’t need to write out a list to figure out the median. We can look at the picture.
Remember, the median is the middle number. That means that you will have the same amount of dots to the left of the median as you do to the right of the median. (50% to the left of the median and 50% to the right of the median)
How do you determine the median in a dot plot?
Median
One strategy that you can use is to…
Mark out the dots starting from the left and right outsides and working inwards until you have determined the middle number.
Watch this!
Another strategy:
Step 1: Count the dots to see if we have one middle number or two middle numbers. An even amount means 2 middle numbers. An odd amount means just 1 middle number. In this case the number of dots is 15, so just one middle number.
Step 2: Take the number of dots and divide by /2 = 7.5 The middle number (median) will be the 8th position.

30. RANGE
The range describes how spread out (or varied) the data is.
Remember… to find the range: is biggest minus smallest.
Since the data on this dot plot is ordered, it is pretty clear that the largest number is 4 and the smallest is 0.
Range: 4 – 0 = 4

31. Let’s look at another example…

32. MEAN
(5) 10 = = =𝟐.𝟕
MEAN
If you split out the 27 trips to the gym evenly between
the 10 people…
They would each have 2.7 trips.

33. I think he might need to cut back on some of his trips to the gym!
MEDIAN
This plot is ordered from least to greatest. We, again don’t need to write anything out.
Even though this plot is fairly symmetrical, it is not perfectly symmetrical. We can still use the same approach as we did on the last example.
Watch this!
Another strategy:
Step 1: Count the dots to see if we have one middle number or two middle numbers. An even amount means two middle numbers. An odd amount means just one middle number. In this case the number of dots is 10, so there are two middle numbers.
Step 2: Take the number of dots and divide by /2 = 5 The two middle numbers will be the 8th position and 9th position. Average the two middle numbers if they are different.
I think he might need to cut back on some of his trips to the gym!
Median = 3

34. RANGE
5 – 0 = 5

35. You Try! Directions: Find the mean, median and range.
The median is a little challenging.
Hint: There are 14 dots so there will be two middle numbers.
This one is more challenging, but see if they can figure out what to do. They will need to fill in the missing numbers on the number line.
There are an even amount of dots so there will be two middle numbers to have to cope with on the median. The two middle numbers do not fall in the same column like the previous example. They will have to get an average.
Proceed to next two slides for answers.

36. You Try!
ANSWER
MEAN
Split evenly…
All 14 people in the survey would get 3.1 servings of fruit per day!
This one is more challenging, but see if they can figure out what to do. They will need to fill in the missing number line intervals.
There are an even amount of dots so there will be two middle numbers to have to cope with on the median. The two middle numbers do not fall in the same column like the previous example. The mean is not too complicated.
Proceed to next two slides for answers. 1 3 5 7
(5) 14 = = = 3.1

37. You Try! ANSWER MEDIAN 3 + 4 = 7 ÷ 2 = 3.5 Median = 3.5 Note…
Any time you have an even amount of dots, there will be two middle numbers. If the middle numbers are different you have to: Add and divide by 2.
3 + 4 = 7 ÷ 2 = 3.5
It doesn’t matter if they “boxed” the two dot that I did. They could have chosen that top dot under the 4 column.
Students can also us different strategy:
There are 14 dots which means that we have two middle numbers.
14/2 = 7 The two middle numbers will be in the 7th and 8th positions.
Get the average of 3 and 4. 1 3 4 5 7
Median = 3.5

38. You Try!
ANSWER
Range
Discuss the spread of the data.
5 –1 = 4

39. Let’s Get Started…
Section 1: Describing a data distribution by its shape.
Section 2: Describing a data distribution by its center and variation.
Section 3: Finding the mean, median, and range
on a Dot Plot
Section 4: Finding the mean, median, and range on a Histogram

40. HISTOGRAMS
…Frequency…
an interval
Finding the range on a Histogram is pretty straight forward. 90 – 31 = 50
However, when trying to find the center (mean and median), there is a slight issue…
Histograms have intervals. Each interval includes a range of numbers.
Therefore, you can’t determine the exact mean and median. You can only get an estimate.

41. Estimating Maybe the numbers are …32 31 40 31 35 31
Maybe the numbers are …
Since we don’t know the exact numbers inside the intervals,
then the mean and median will not be exact.
We can only identify the interval in which they are located.
We know there are 3 numbers in this interval, but we don’t know exactly which 3 numbers.
We only know that the three numbers are from 31 to 40.
Watch this

42. Looking at the frequencies for each interval…
MEDIAN 8 3
I am not showing how to get the mean because it is a little more complicated. You can explain it if you choose to.
They would have to multiply the middle value in each bar by the height of the bar. Then, add all of those products and divide by the total number of days.
Looking at the frequencies for each interval…
Start from the left and count until you are at the interval that is holding the 15th number.
- There are 3 in the first interval;
- Then 8 more numbers in the second interval;
We find that the 15th position is somewhere in the third interval and… so is the 16th position!
Watch this !
We have 30 numbers in the data set (number of days).
We want the middle number.
30 ÷ 2 = 15
So… we are looking for the 15th and 16th position.
I know there are two middle numbers because 30 is an even number.

43. Closure
1) In which data distribution below is the mean > median? 2) In which is the mean < median? 3) In which is the mean = median? Plot A Plot B Plot C 4) The mean and median measure __________? Center or Variation 5) The range measures ________? Center or Variation 6) What does “measures of variation” refer to? Give an example of a measure of variation. 7) What two types of graphs did we look at today? _________ plots and ________?
Plot C. The data is right skewed, so the mean will be pulled to the right of the median.
Plot A. The data is left skewed, so the mean will be pulled to the left of the median.
Plot B. The data is symmetrical. The mean and median will be equal.
Center
Variation
It refers to how spread out the data (or how varied) it is.
Dot plots and histograms.

44. Closure 8 Is it possible to find the mean and median on a dot plot?
Is it possible to find the mean and median on a histogram?
9) If you have 8 numbers in a data set, how many middle numbers are there?
In which position(s) will they be located?
10) If there are 9 numbers in a data set, how many middle numbers are there?
In which position will the median be?
Yes on the dot plot; Only an estimate on a histogram because of the intervals.
If 8 numbers there will be two middle numbers. To find the position: 8/2 = 4 so will be in 4th and 5th positions.
If 9 numbers there will be only one middle number. 9/2 = 4.5 so will be at the 5 position