1. Common Core Math I Unit 2: One-Variable Statistics Measures of Center and Spread

2. Think About the Situation
Discuss the following with your partner or group. Write your answers on your own paper. Be prepared to share your answers with the class.
What do you think is the typical number of letters in the full names (first and last) of your classmates?
What data do you need to collect and how would you collect it?
How would you organize and represent your data?
If a new student joined your class today, how might you use your results to predict the length of that student’s name?
This is the warm up for day 4. It is also printed in the handouts for today. Discuss answers with whole class.
Possible answers/additional questions:
What data do you need to collect and how would you collect it?
Ask each group to share their ideas. There is no one right answer here.
Need a list of students names. Get it from the teacher. Poll the class. Ask everyone to write their first and last names on the board. Ask everyone to write the number of letters in their first and last names on the board.
How would you organize and represent your data?
Look for multiple answers to cover what has been covered so far: make a frequency distribution table, dot plot, histogram. Ask: What intervals would you use?
If a new student joined your class today, how might you use your results to predict the length of that student’s name?
Again, expect multiple ideas here, but focus on the center. The center of the distribution describes the typical value, so that is what we can use to predict the length of the new student’s name.

3. Investigation 1: Dotplots vs. Histograms
Have students complete Investigation 1 from handout. Be sure to discuss as a class.
Possible answers/additional questions:
Describe the distribution of the data in context (shape, center, spread, outliers).
The distribution of name lengths of students in Ms. Jackson’s class is fairly symmetrical (or some may say slightly skewed rleft) with center around 13 letters. The length of the names varies from 9 letters to 17 letters. There are no apparent outliers. (Some students may say that 9 and/or 17 are outliers because of the gap between the main cluster and these values. This is OK, as long as they justify why they think they are outliers. If there is disagreement on this, that is a good thing because it will lead to the need for a process to identify outliers, which will come later in the unit. If no student mentions outliers, ask “Isn’t 17 an outlier?”)
How are the two graphs alike? How are they different?
In this case, their shapes look exactly alike because the histogram is using interval length of 1. How would this change if we used intervals of 2 or 5? The height of the stacks/bars shows the frequency of each name length.
The dot plot is using a dot to represent each data value but the histogram uses bars. The count is given by the height of the bar. There is a y-axis that shows the frequency.
How can you use each graph to determine the total number of letters in all the names?
Some students might misunderstand this question and give the total number of students, not letters. For example: I could count up all of the dots in the dot plot. In the histogram, I could find the frequency of each bar and add up the frequencies. If this misconception doesn’t come up, bring it up and have students explain why this is an incorrect interpretation.
To find the total number of letters, we need to multiply each value for the number of letters times the frequency of for that value and then add them up. In symbol form,  xf , where x represents a data value and f represents the frequency. This leads into the formula for weighted mean, which is x-bar = ( xf )/,  f . It is not necessary to formalize this with students, but may be helpful to make the connection when you talk later about finding the mean when given a frequency distribution (and not a list of data values). Some students may “un-do” the graph and make a list of values and then add them up. This is not wrong, just inefficient. You could have a discussion on which of the students methods is most efficient, but be careful not to force one method over another!
Cassandra Smithson said, “My name has the most letters, but the bar that shows my name length is one of the shortest on the graph. Why?” How would you answer this question?
The bar heights represent how many students have that name length, not the length of the name itself. So 17 has a bar height of one because only one student has that name length – Cassandra.

4. Describing Data Two ways to describe data: Graphically Numerically
Dot plot
Histogram
Boxplot
Numerically
Measures of Center
Measures of Spread
Use this slide to remind students of where they are in the study of one-variable statistics. This next lesson deals with describing data numerically.

5. Describing Data Numerically
Measures of Center – mean, median
Measures of Spread – range, interquartile range, standard deviation
Again, just to remind students where they are in the study of one-variable stats. So today’s focus is on the median.
S-ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

6. What is the typical value?
Measures of Center
What is the typical value?
These definitions are a little more student friendly than the ones in C-MAPP and are OK to use. The ones in C-MAPP are more precise.

7. Investigation 2: Finding the Median
Have students complete Investigation 2 from the handouts. Be sure to discuss as a class.
Possible answers/additional questions:
Fold the strip in half.
What number does the crease fall on? 13
How many name lengths are below this number? 11
How many name lengths are above this number? 11
The median is the midpoint of the data set. The same number of data values fall above and below this value. What is the median of this data set? 13 letters
Suppose Cassandra Smithson moves to another school. The class now has 22 students. On your strip of squares, cut off the square that corresponds to Cassandra’s name length. Fold the strip in half.
What number does the crease fall on? It falls exactly between two 13s.
How many name lengths are below this number? 11
How many name lengths are above this number? 11
What is the median of this data set? 13
Discuss a scenario with students where there are 22 students in a class and the crease falls between the numbers 11 and 13. What would the median be? (12) We want students to see the median as the midpoint of the data.
There are 15 students in a class. Use the information about the class’s name lengths below to answer the questions.
Most common name length: 12 letters
Median: 12 letters
Range: The data vary from 8 letters to 16 letters.
Find a possible set of name lengths for this class. Describe the process you used. Some students may use guess and check. Others will build the data set so that there is the same number of values (7) above and below 12. Get a variety of responses and processes.
Make a dot plot to display the data.
Compare your graph with the graphs of your classmates. How are they alike? How are they different? Alike: they all fit the three conditions given. (If they don’t, have students give ideas to “fix” the graphs to meet all three conditions.) Different: different shapes, different clusters, gaps, etc. maybe 16 is an outlier in one of the graphs with the rest of the values clumped around Give some unusual “wrong” answers and have students analyze. Display a graph where all 15 values are 12. Is this possible? (The median is 12 and 12 is the most common value, but there is no variability from 8 to 16 letters.) Also, show a graph where 7 values are 8, one is 12, and 7 values are 16. Is this possible? (The median is 12 and the data varies from 8 to 16 letters, but 12 is not the most common number of letters.)

8. Investigation 3: Experimenting With the Median
Name
Number
of Letters
Pete Thomas 10
Shaquana Smith
Stewart Hughes
Huang Mi
Richard Lewis
Virginia Bates
Angel Mendoza
Mary Wall
Danielle Duncan
Will Jones
Ana Romero
Jana Turner
Huang Mi 7
letters
Have students complete Investigation 3: Experimenting with the Median on the handout.
Possible answers/additional questions:
Order the cards from shortest name length to longest name length, and identify the median of the data. What is the median? 11
1. Remove two names from the original data set so that:
the median stays the same. What names did you remove? Answers will vary, but they should remove one from the lower half and one from the upper half.
the median increases. What names did you remove? Answers will vary,
but they should remove two numbers from the lower half.
the median decreases. What names did you remove? Answers will vary,
but they should remove two numbers from the upper half.
2. Now, add two names to the original data set so that:
the median stays the same. What names did you add? Answers will vary,
but they should one number to the lower half and one to the upper half.
the median increases. What names did you add? Answers will vary,
but they should add two numbers from the upper half.
the median decreases. What names did you add? Answers will vary,
3. How does the median of the original data set change if you add
a name with 16 letters? The median will be 12.
a name with 4 letters? The median will be 10.
the name William Arthur Philip Louis Mountbatten-Windsor (a.k.a. Prince William) to the list? The median will be 12.

9. Slides that follow are for day 5

10. Describing Data Numerically
Measures of Center – mean, median
Measures of Spread – range, interquartile range, standard deviation
Again, just to remind students where they are in the study of one-variable stats. Today’s focus is on the mean.
S-ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

11. Think About the Situation
Use cubes to make stacks representing each household. Use the stacks to answer the following questions. What is the median of these data?
Paul
Arlene
This is also printed in the handouts for today. Questions appear on this slide and following slides.
What is the median of these data? The median is 3.5 people per household.
Ruth
Ossie
Gary
Leon
Adapted from Data About Us, Connected Mathematics 2, Grade 6

12. Finding the Mean Make stacks all the same height by moving cubes.
Make the stacks all the same height by moving the cubes.
Ossie
Paul
Gary
Leon
Ruth
Arlene
Adapted from Data About Us, Connected Mathematics 2, Grade 6

13. Finding the Mean How many cubes are in each stack?
By leveling out the stacks to make them equal height, you have found the average, or mean, number of people in a household. What is the mean number of people per household?
How many cubes are in each stack? 4
By leveling out the stacks to make them equal height, you have found the average, or mean, number of people in a household. What is the mean number of people per household? 4 people per household
Ruth
Ossie
Gary
Paul
Leon
Arlene
Adapted from Data About Us , Connected Mathematics 2, Grade 6

14. Investigation 1: Finding the Mean
How many people are in the six households altogether? Explain.
What is the mean number of people per household for this group? Explain how you got that number.
How does the mean for this group compare to the mean of the first group?
What are some ways to determine the mean number of a set of data other than using cubes? How do these methods relate to the method of using the cubes?
Student Name
Number of People
in Household
Reggie 6
Tara 4
Brendan 3
Felix
Hector
Tonisha
Have students complete Investigation 1: Finding the Mean.
Possible answers/additional discussion questions:
How many people are in the six households altogether? Explain. I added up all of the households to get 24 people total.
What is the mean number of people per household for this group? Explain how you got that number. I leveled out the heights for the stacks of cubes and got 4 cubes in each stack, so 4 is the average number of people per household.
How does the mean for this group compare to the mean of the first group? It is the same as the first group.
What are some ways to determine the mean number of a set of data other than using cubes? How do these methods relate to the method of using the cubes? Students may mention the formula. Other ways are to think about the mean as a balance point or as a fair share. (see following slides on mean as fair share and mean as balance point)

15. Mean as a Fair Share Mary – 7 cookies Tom – 6 cookies
Julio – 2 cookies
Mary, Tom, and Julio are good friends. They got together and decorated gingerbread men cookies. Mary decorated 7, Tom 6, and Julio 2. (he got there late) Now they are ready to eat them! Since they are such good friends, they decide to share them equally, even though Mary decorated more than Tom or Julio. How many cookies should each of them get? What would be a fair share? Well, we have 3 people, so let’s rearrange until we have 3 equal groups of cookies. (Slide #16) How many cookies does each get? Right, five. How many total cookies are there? Notice that if we divide 15 into 3 equal groups, we will get 5.
Mathematicians call what we just found the “mean.” Another word for this is the arithmetic average. Have you heard these terms before? (they should have – this was covered in middle school). How do we calculate the mean? (Slide #17). Right, we find the total of our values and divide by the number of values. In this case, we have = 15 total, and we divide by 3 because there are three values – 7, 6, and 2 – giving the result of 5. So the mean of 7, 6, and 2 is 5. We could say that each person gets five cookies on average. So calculating the mean is a way of dividing things up so that each person gets a fair, or equal, share.

16. Mean as a Fair Share
How many cookies will each child get if they each get an equal share?
How many cookies does each get? Right, five. How many total cookies are there? Notice that if we divide 15 into 3 equal groups, we will get 5.
We could say that each person gets five cookies on average. So calculating the mean is a way of dividing things up so that each person gets a fair, or equal, share.

17. Mean as a Balance Point 2 4 3 5 6 7
Now let’s look at the mean in another way. I have the numbers 2, 6, and 7 graphed on a number line. If I wanted to take that number line and find a point where I could put my finger and it would balance like a see-saw, it would be at the mean of 5.

18. Mean as a Balance Point
Think about it this way. On a see-saw, the fulcrum, or pivot point, is usually in the middle. So if two kids of equal weight get on, they will balance each other out and the see-saw will be horizontal.

19. Mean as a Balance Point
If the two kids aren’t equal weight, then the heavier one will tip the see-saw down in their direction.

20. Mean as a Balance Point
We know from physics that if we move the fulcrum closer to the heavier weight, we can get the two sides to balance again. The tip of the fulcrum points to the “balance point” – where the fulcrum needs to be to keep the see-saw level.

21. Mean as a Balance Point
So let’s think of our number line as representing the see-saw. The numbers of 2, 6, and 7 represent people sitting on it. If I have two people sitting on the right side, the see-saw will tip towards them.

22. Mean as a Balance Point
So I have to move the fulcrum over closer to them in order to find the balance point.
So the mean of 2, 6, and 7 is 5.

23. Mean as a Balance Point
Let’s look at finding the balance point for 1, 2, 6, 7.
This time they are already balanced! So the mean is 4.

24. Mean as Balance Point

25. Mean as a Balance Point
How about 1, 3, 6, 7? Is this balanced? (Hopefully from their experience with see-saws, they know that if a person moves in a closer distance, then they don’t balance with someone of equal weight who is out a further distance). No. Which way will it tip? Right, towards the 6 and 7. (The whole fulcrum/length of lever principle) So what will the balance point be? If we find the mean of 1, 3, 6, and the sum of these numbers is 17, and there are 4 numbers, so 17 divided by or 4 ¼. So we have to slide the fulcrum over just a little to reach our balance point.

26. Thinking About the Situation
Answers to questions/additional questions:
There are six households in each group.
There are = 24 people in Group A and = 24 people in Group B.
The number of people divided by the number of households tells us the number of people per household, which is the mean.
This leads into finalizing the formula for mean (next slide)

27. The Formula Discuss what symbols mean!
x-bar is the symbol for the mean of the
is the capital Greek letter sigma, which stands for “sum.”
x is an individual data value
n is the number of data values.
Note: Students do NOT need to memorize this formula. They need to know, use, and be able to describe the process for finding the mean.

28. Investigation 2: Data with the Same Mean
Find two new data sets for six households that each have a mean of 4 people per household. Use cubes to show each data set. Then make dotplots from the cubes.
Find two different data sets for seven households that each has a mean of 4 people per household. Use cubes to show each data set. Then make dotplots from the cubes.
A group of seven students find that they have a mean of 3 people per household. Find a data set that fits this description Then make a dot plot for this data.
A group of six students has a mean of 3.5 people per household. Find a data set that fits this description Then make a dot plot for this data.
How can the mean be 3 ½ people when “half” a person does not exist?
How can you predict when the mean number of people per household will not be a a whole number?
Answers to questions/additional questions for discussion:
Find two new data sets for six households that each have a mean of 4 people per household. Use cubes to show each data set. Then make dotplots from the cubes. Have students share their different dotplots. Discuss similarities and differences. All should have 24 total people and for each dot plot, 4 is the “balance point.” If it doesn’t come up, show a dot plot where all values are 4 and another where 3 values are 2 and 3 values are 6.
Find two different data sets for seven households that each has a mean of 4 people per household. Use cubes to show each data set. Then make dotplots from the cubes. Same as above. All should have 28 total people.
A group of seven students find that they have a mean of 3 people per household. Find a data set that fits this description. Then make a dot plot for this data. There should be 21 total people represented.
A group of six students has a mean of 3.5 people per household. Find a data set that fits this description. Then make a dot plot for this data. There should be 21 total people.
How can the mean be 3 ½ people when “half” a person does not exist? Since we are dividing, its possible to get a value that is not a part of the data set. When we interpret, we need to take the context of the data into account. In this case ½ a person doesn’t make sense, but when we interpret this value, it tells us that the typical household for this group has 3-4 people in it.
How can you predict when the mean number of people per household will not be a a whole number? When the number of households is not a factor of the number of total people, it will not divide evenly to give us a whole number.

29. Investigation 3: Using the Mean

30. Investigation 3: Using the Mean
Find the following:
the total number of students
the total number of movies watched
the mean number of movies watched
A new value is added for Carlos, who was home last month with a broken leg. He watched 31 movies.
How does the new value change the distribution on the histogram?
Is this new value an outlier? Explain.
What is the mean of the data now?
Compare the mean from question 1 to the new mean. What do you notice? Explain.
Does this mean accurately describe the data? Explain.
Answers to questions/additional discussion questions:
a) 10 90
9 movies per student
a) There would be an additional bar between 30 and 35 with a frequency of 1.
This would make the distribution appear to be skewed to the right.
Yes, there is a large gap between 31 and the rest of the data.
121/11 = 11 movies per student
It is higher than the previous mean by 2.
Take students opinions on this. Some may say yes, some no. Note during the class discussion that 5 of the 11 students have watched 11 movies or more. If time: let’s look at the median number of movies. What is the median number of movies watched both before Carlos is added (6.5) and after Carlos is added (7)? Which value seems to describe the data better in each case – the mean or the median? This discussion lays the groundwork for the next part of the lesson on mean vs. median.

31. Data for eight more students is added.
Add these values to the list in your calculator. How do these values change the distribution on the histogram?
Are any of these new values outliers?
What is the mean of the data now?
The distribution is more clustered to the left, so it now appears skewed right.
The new values are not outliers – they all fall in the bottom cluster.
The mean is 7.9, or approximately 8, movies per student.

32. How do I know which measure of central tendency to use?

33. Investigation 4: Mean vs. Median
The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft 5 in, 5 ft 3 in, and 5 ft 7 in. A student transfers to Washington High and joins the basketball team. Her height is 6 ft 10in.
Discuss and solve in your groups!
Let’s talk about the mean vs. the median. How do you know which one to use? Consider this situation:
Read the problem, then give groups time to answer the questions on the worksheet. Discuss answers.
What is the mean height of the team before the new player transfers in? (65.9 in.) What is the median height? (66 in.)
What is the mean height after the new player transfers? (67.9 in.) What is the median height? (66.5 in.)
What effect does her height have on the team’s measures of center? (The mean increased by 2 in. and the median increased by .5 in.)
How many players are taller than the new mean team height? (2) How many players are taller than the new median team height? (4)
Which measure of center more accurately describes the team’s typical height? Explain. (The median gives a more accurate description of the team’s typical height. Half of the players are taller than the median (and half shorter) but only two players are taller than the mean. Using the mean would lead someone to conclude that the team is taller than they really are.)

34. Mean vs. Median http://www.stat.tamu.edu/~west/ph/meanmedian.html
Let’s look at how outliers affect the placement of the mean and the median relative to the distribution. Use the applet to explore the effect of outliers on the mean and the median.

35. 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.

36. Ticket out the Door
What happens to the mean of a data set when you add one or more data values that are outliers? Explain.
What happens to the mean of a data set when you add values that cluster near one end of the original data set? Explain.
Explain why you think these changes might occur.

37. Slides that follow are for day 6

38. Describing Data Two ways to describe data: Graphically Numerically
Dot plot
Histogram
Boxplot
Numerically
Measures of Center: Median and Mean
Measures of Spread
Use this slide to remind students of where they are in the study of one-variable statistics. This lesson continues our study of describing data numerically. Now we turn to measures of spread.

39. Measures of Spread How much do values typically vary from the center?
Range
Interquartile Range (IQR)
Standard Deviation
Today’s focus will be on standard deviation.

40. Thinking about the Situation
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 86 89 84
Anna 80 99 73 88
Let students discuss this in their groups and make a decision. Have them share out their ideas.
Answers:
What is the mean score for each student? 85 points
Based on the mean, who is the best student? Let them share their opinions. This may end up running into the third question. If it doesn’t come up, say “They all have the same mean, so they are equal as students.” What do they think?
If asked to select one student, who would you pick as the best student? Explain. Accept their explanations without judgment. Some may look at Johnny’s increasing trend over time. Some may look at Will’s consistency. Some may note Anna’s one low grade. Be sure that arguments for each student being the “best” come out, even if you have to be the one who makes them. The point is, just using the mean to describe each student is not enough. I think that we can all agree that they are not “equal” in their test performance. We need more information than just the typical test score. One thing to look at is how consistent each student is, and measures of spread will give us that information.

41. 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. In this lesson we will talk about the mean absolute deviation and how it is related to the standard deviation. What does the word “deviation” mean? (1. The action of departing from an established course or accepted standard The amount that a single measurement differs from a fixed value such as the mean.)

42. Investigation 1: Deviation from the Mean
Score
Mean
Deviation from
the Mean
Anna
Test 1
Test 2
Test 3
Test 4
Have students work through Investigation 1: Deviation from the Mean
2) The sum of the deviations for each student is zero. The mean is the balance point, so the distances between the mean and each point balance in either direction. Go through next slides to solidify this concept.

43. 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

44. Houston, we have a problem!
Measure of Spread
Sum of deviations =
(-4)+(-3)+(-1)+(+5)+(+3)
Average of the deviations= = 0
An average deviation of zero means that there is no variability!
Remember that we want to have some kind of measure of each student’s consistency. So let’s try to find a way to measure how much each student’s test grades vary from their average test grade. 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 to find the average we would add up the deviations and divide by the number of data values that we have.
Houston, we have a problem!

45. 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
Ask students to give their ideas on fixing the problem of the deviations adding up to zero. Although they may not state using the “absolute value”, they may state the concept by suggesting that we ignore the signs and just add up the numbers.

46. 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.

47. How can we fix our problem?
Investigation 2: Mean Absolute Deviation
Deviation from the Mean
Absolute Deviation from the Mean
Johnny
Test 1
Test 2
Test 3
Test 4
Will
Anna
Have them complete Investigation 2. Discuss.
2) The MAD values are: Johnny 11.5, Will 2.5, and Anna 8.75.
3) The MAD tells us the average deviation from the mean for each student, telling us how much their test scores typically vary from their average test score. Will’s MAD is the lowest, indicating that his test scores are the least spread out.
4) Johnny’s test scores typically fall within points of his average score of 85. So Johnny’s scores typically vary anywhere from 74 to 96.

48. 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.

49. Back to Johnny, Will and Anna . . .
Investigation 3: Calculating the Standard Deviation
Student
Test 1
Test 2
Test 3
Test 4
Johnny 65 82 93
100
Will 86 89 84
Anna 80 99 73 88
Have students complete Investigation 3. You may want to walk them through Johnny’s data and then have them do Will’s and Anna’s in their groups and answer the questions.
Discussion Questions:
Why is the sum of the third column always equal to zero? The sum of the deviations from the mean is always zero because the mean is the balance point of the data.
Translate into words: The sum of the squared deviations from the mean. Or from inside parentheses out, subtract the mean from each value, square these differences and add them up.
Interpret Anna’s standard deviation in context. Anna’s test grades typically vary by about ten points from her average score of 85. So her test scores typically vary from about 75 to 95.

50. 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 86 89 84
2.2
Anna 80 99 73 88
9.7
Let students discuss this in their groups and make a decision. Have them share out their final verdict.

51. 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 (last page of handout).

52. s vs.  What’s the difference?
The sample standard deviation is used to estimate the population standard deviation. If we divide only by n, the sample standard deviation estimate tends to be too low, so we correct that by dividing by a smaller number.
Be sure that students know that the calculator gives both Sx and x when using 1-Var Stats and they should choose which one to use. If not sure, have them pick the larger of the two (the population standard deviation sigma) to be on the safe side.
population
sample