6.SP.A.3Common CoreMathStatistics and ProbabilityGrade 6
6.SP.A.3: Measures of Center and Measures of Variation
In plain English: 6.SP.A.3 is the Common Core grade 6 math standard that asks students to recognize two kinds of summary numbers for a numerical data set (a list of number answers). A measure of center, such as the mean (the fair share) or the median (the middle value), uses one number to stand for all the values. A measure of variation, such as the range (largest minus smallest), uses one number to tell how much the values differ.
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.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Develop understanding of statistical variability. Also written as 6.SP.3 · Official standard
A numerical data set is a list of number answers to one question, such as the number of cans each student brought to a food drive. Listing every value takes a lot of space, so statisticians use summary numbers. This standard asks students to recognize two kinds. A measure of center is one number that stands for all the values in the set, such as the mean (the fair share: add the values and divide by how many there are) or the median (the middle value when the values are in order). A measure of variation is one number that tells how much the values differ from each other, such as the range (largest value minus smallest value).
Students level towers of cubes to see the mean as a fair share, time themselves estimating one minute, and sort statements by the kind of number they use. They also meet the mean absolute deviation (the average distance of the values from the mean) on small, friendly data sets, as a second measure of variation. The key idea is that one number cannot do both jobs: two data sets can have the same center and very different variation. Using these measures to summarize and compare full data sets comes next, in 6.SP.B.5. All data on this page are invented for teaching and labeled as invented.
Learning Objectives
By the end of this lesson, students will be able to:
Explain that a measure of center, such as the mean or the median, is one number that summarizes all the values of a data set
Find the mean as a fair share and the median as the middle value of a small data set
Explain that a measure of variation, such as the range or the mean absolute deviation, is one number that tells how much the values vary
Show with an example that two data sets can have the same center but different variation
Decide whether a statement about data uses a measure of center or a measure of variation
Prior Knowledge Required
Students should already be comfortable with:
Sharing a total equally among several people, as in redistributing liquid in beakers so each holds the same amount 5.MD.B.2
Recognizing statistical questions, which expect answers that vary 6.SP.A.1
Adding, subtracting and dividing multi-digit numbers and decimals 6.NS.B.26.NS.B.3
Finding the distance between two numbers on a number line
Show the prompt and give students two minutes to answer in pairs:
Warm-Up Prompt
"Two basketball players each scored 60 points in their last 5 games. Player 1 scored 12 points every game. Player 2 scored 2, 25, 4, 21 and 8. If you could tell a coach only ONE number about each player, what would it be? Is one number enough to show how different the two players are?"
Many pairs will say "12 points per game" for both players: 60 ÷ 5 = 12. Point out that this one number is the same for both players, yet the players are very different. Player 1 is steady; Player 2 goes up and down a lot. Tell students that today they will learn two kinds of single numbers: one that tells a typical value (the center) and one that tells how much the values change (the variation).
Direct Instruction20 minutes
Part 1: A measure of center summarizes all the values. The mean is the fair share. Add all the values, then divide by the number of values. Diagram 1 shows why: if students move cans from the tall bars to the short bars until every bar is the same height, each bar ends at the mean. Every value helps decide the mean, so the mean summarizes all the values with one number. The median is the middle value after the values are put in order (or halfway between the two middle values). It also stands for the whole set with one number.
The mean as a fair share
Invented data: cans collected for a food drive by 5 students: 12, 15, 9, 18, 16 (Diagram 1). If the students share the work equally, how many cans is that for each?
Equation: Total: 12 + 15 + 9 + 18 + 16 = 70 cans. Mean: 70 ÷ 5 = 14 cans per student. The median, the middle of 9, 12, 15, 16, 18, is 15 cans
One center for two very different sets
Invented data: stickers per page in two sticker books with 4 pages each. Book A: 6, 6, 6, 6. Book B: 1, 5, 7, 11.
Equation: Both means are 24 ÷ 4 = 6 stickers. Book A's range is 6 - 6 = 0, and Book B's range is 11 - 1 = 10, so the center alone hides the difference
Same mean, different variation
Invented data: daily high temperatures (°F) for 5 days. Town A: 69, 71, 72, 73, 75. Town B: 58, 65, 74, 79, 84 (Diagram 2).
Equation: Both means are 360 ÷ 5 = 72 °F. Range: Town A 75 - 69 = 6 °F, Town B 84 - 58 = 26 °F. Town B's temperatures vary much more
Mean absolute deviation
Invented data: laps swum in 10 minutes by 4 swimmers: 5, 8, 9, 14. How far, on average, is each swimmer from the mean?
Equation: Mean: 36 ÷ 4 = 9 laps. Distances from 9: 4, 1, 0, 5. Mean absolute deviation: (4 + 1 + 0 + 5) ÷ 4 = 2.5 laps
Part 2: A measure of variation describes how the values vary. The range is the largest value minus the smallest value. It is one number that tells how far apart the ends of the data are. The mean absolute deviation (MAD) is the average distance of the values from the mean: find each value's distance from the mean, add the distances and divide by the number of values. A distance is never negative, so a value 4 below the mean and a value 4 above it both have a distance of 4. Diagram 2 writes each distance above its dot.
A variation of 0: in Book A, every page has 6 stickers, so every distance from the mean is 0. The range and the MAD are both 0: the values do not vary at all.
Bigger variation, bigger number: Town B's distances from 72 are 14, 7, 2, 7 and 12. They add to 42, so its MAD is 42 ÷ 5 = 8.4 °F. Town A's distances are 3, 1, 0, 1 and 3, so its MAD is 8 ÷ 5 = 1.6 °F.
Two jobs, two numbers: to describe a data set with numbers, give one measure of center and one measure of variation, with units.
Guided Practice15 minutes
Pairs work through the data set below together. They find one measure of center and two measures of variation, then write one sentence about what each number tells.
Invented data: minutes 6 students read on Saturday.
Student
1
2
3
4
5
6
Minutes
20
30
40
25
35
30
Answers: the total is 180 minutes, so the mean is 180 ÷ 6 = 30 minutes. The range is 40 - 20 = 20 minutes. The distances from 30 are 10, 0, 10, 5, 5 and 0, which add to 30, so the MAD is 30 ÷ 6 = 5 minutes. Sample sentences: "A typical student read about 30 minutes." "On average, a student's time was 5 minutes away from 30." Listen for pairs who divide the total by the largest value instead of by the number of values, and for pairs who forget the zero distances when they add.
Independent Practice10 minutes
Students work alone. (1) Invented data: hours worked last month by 5 volunteers at an animal shelter: 3, 4, 4, 5, 9. Find the mean and the median. (Total 25, so the mean is 25 ÷ 5 = 5 hours; the median is 4 hours.) (2) The volunteer who worked 9 hours actually worked 24 hours. Find the new mean and the new median, and explain what changed. (Total 40, so the new mean is 8 hours; the median is still 4 hours. The mean uses every value, so changing one value changes it; the median depends only on the middle value.) (3) Find the range before and after the change. (9 - 3 = 6 hours before, 24 - 3 = 21 hours after.)
Closure5 minutes
Exit ticket: invented data for how many minutes late the school bus was on 4 days: 3, 5, 6, 10. (1) Find the mean. (2) Find the MAD. (3) Which of your two numbers tells a typical value, and which tells how much the lateness changes from day to day? (Mean 24 ÷ 4 = 6 minutes. Distances 3, 1, 0, 4, so the MAD is 8 ÷ 4 = 2 minutes. The mean is the measure of center; the MAD is the measure of variation.)
Differentiation Strategies
For Struggling Students
Let students keep using cube towers or drawings of bars to find each mean before moving to adding and dividing
Give a three-column table (value, mean, distance) for every MAD problem, with the mean already written in
Post an anchor chart with two columns, "Center: one typical value" and "Variation: how much the values differ", and have students place each new measure in a column
For Advanced Students
Ask students to write two data sets of 5 values with the same mean and the same range but different MADs
Ask what happens to the mean and to the MAD when you add 10 to every value in a data set, and why
Ask students to find a data set whose mean and median are far apart, and to say which one they would report as the typical value
Assessment Guidance
What to Look For
Students should be able to say which job a number does: "this is a typical value" (center) or "this tells how spread out the values are" (variation). When they find a mean, check that they divide the total by the number of values. When they find a MAD, check that every distance is positive and that they divide by the number of values. A strong answer gives a center and a variation together, with units, and explains why one number is not enough.
02
Classroom Activities
3 Activities
1
Level the Towers
15 minGroups of 3-4
Each group gets 2 tower cards and a bag of linking cubes. Groups build 5 towers with the heights on a card, then move cubes one at a time until all 5 towers are the same height. The level height is the mean, a single number that summarizes all 5 towers.
Tower Cards (4 cards; each group gets one pair)
Card A: towers of 3, 5, 6, 8 and 8 cubes
Card B: towers of 2, 4, 4, 9 and 11 cubes
Card C: towers of 5, 7, 7, 8 and 8 cubes
Card D: towers of 1, 3, 7, 10 and 14 cubes
Pair A with B, and C with D. Each pair of cards has the same total (30 cubes for A and B, 35 cubes for C and D), so both cards in a pair level to the same height.
Procedure
Build the towers from the first card and record the heights
Move cubes from taller towers to shorter towers until all are level; record the level height
Check by adding the heights and dividing by 5
Count how many cubes you had to move, then repeat with the second card
Discussion Questions
Cards A and B level at the same height. What is it? (6 cubes: 30 ÷ 5 = 6)
Which card of each pair needed more cubes moved? (Card B: 8 cubes, compared with 4 for Card A; Card D: 10 cubes, compared with 2 for Card C)
The level height is the same for Cards A and B. What single number could show that Card B's towers were more uneven? (The range: 5 cubes for Card A and 9 for Card B)
If you add one more tower of exactly the level height, does the level height change? Why?
Modification for Distance Learning
Students use coins, pasta pieces or drawn squares on grid paper instead of cubes, and share a photo of the towers before and after leveling.
2
Guess One Minute
20 minPairs, then whole class
This activity answers the statistical question "How close can students in our class come to guessing one minute without a clock?" One partner closes his or her eyes and raises a hand when he or she thinks 60 seconds have passed; the other partner times it. The class then summarizes the results with a measure of center and a measure of variation.
Procedure
The timer says "Go" and starts the stopwatch; the guesser raises a hand at what feels like one minute
The timer records the time to the nearest second; partners switch roles
Each pair adds its two times to a class list on the board
As a class, find the mean and the range; each pair then finds the distance of its own times from the class mean
Sample Results (invented, 10 students)
42, 48, 51, 55, 57, 58, 62, 64, 70, 83 seconds. The total is 590 seconds, so the mean is 59 seconds. The range is 83 - 42 = 41 seconds. The distances from 59 are 17, 11, 8, 4, 2, 1, 3, 5, 11 and 24, so the MAD is 86 ÷ 10 = 8.6 seconds.
Discussion Questions
In the sample, the mean is close to 60 seconds. Does that mean most students guessed well? (No: only 3 of the 10 guesses are within 3 seconds of 60, and one guess is 23 seconds too long)
Which number tells how good the class is at guessing one minute on average? Which tells how much the guesses differ?
If the class practiced and tried again, which number would you expect to get smaller?
Variation: Guess 30 Seconds
Repeat with a target of 30 seconds and compare the two MADs. Is it easier to be consistent with a shorter time?
3
Center or Variation? Statement Sort
15 minPairs
Pairs get 8 statement cards about invented data. They sort each card into one of two piles, "measure of center" or "measure of variation", and name the measure it uses when they can.
Statement Cards (8 cards)
Card 1: "A typical backpack in our class weighs 11 pounds."
Card 2: "The heaviest backpack weighs 9 pounds more than the lightest."
Card 3: "If all the pizza slices were shared equally, each student would get 3."
Card 4: "On average, each runner's time is 40 seconds away from the mean time."
Card 5: "Half the plants are shorter than 18 centimeters and half are taller."
Card 6: "The phone battery lasted between 10 and 14 hours every day this week."
Card 7: "The daily high temperatures this week were all within 2 degrees of 65 °F."
Card 8: "Our class raised $4.50 per student for the field trip."
Answer Key
Center: Cards 1, 3, 5 and 8 (a typical value, the mean as a fair share, the median, a fair share). Variation: Cards 2, 4, 6 and 7 (the range, the MAD, a range of 4 hours, all values within 2 degrees of a center). Card 7 mentions a center too, but its main message is how little the values vary.
Discussion Questions
Card 6 gives two numbers, 10 and 14. What single number of variation do they give? (A range of 4 hours)
Rewrite Card 1 so that it uses a measure of variation instead.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Mean as a Fair Share
Bars show the invented food drive data from the first worked example, drawn to scale: 12, 15, 9, 18 and 16 cans. The dashed line is the mean, 14 cans. The parts of the bars above the line (1, 4 and 2 cans) exactly fill the gaps below the line (2 and 5 cans), which is why the mean is a fair share of all the values.
Diagram 2: Same Mean, Different Variation
Two invented data sets with the same mean of 72 °F, drawn to scale as dot plots. The number above each dot is its distance from the mean. Town A's distances are small, so its range (6 °F) and MAD (1.6 °F) are small. Town B's distances are large, so its range (26 °F) and MAD (8.4 °F) are large.
04
Homework Assignment
~30 min
6.SP.A.3 Homework: One Number for Center, One Number for Variation
Directions: All data sets are invented. Show your work, write units with every answer, and say in words what each number tells about the data.
Part 1: Measures of Center (Problems 1-2)
Six students recorded the pages they read yesterday: 12, 18, 20, 20, 25, 31. Find the mean and the median. Write one sentence that uses one of them to describe the whole group.
Four friends earned $18, $25, $30 and $27 raking leaves. They put the money together and share it equally. How much does each friend get? Explain why this amount is the mean of the four earnings.
Part 2: Measures of Variation (Problems 3-4)
Two archers each shot 5 arrows, scoring from 1 to 10 points per arrow. Kim: 7, 8, 8, 9, 8. Lee: 4, 10, 6, 10, 10. Find the mean, the range and the MAD for each archer. Which archer is more consistent? Which numbers show it?
A bakery packs 5 bags of rolls, and each bag has exactly 12 rolls. (a) What are the mean, the range and the MAD of the number of rolls per bag? (b) One bag is repacked with 17 rolls and another with 7 rolls. Does the mean change? Does the range change? Explain.
Part 3: Choosing and Explaining (Problems 5-6)
For each statement, write "center" or "variation" and name the measure if you can: (a) "The typical wait at the lunch line is 6 minutes." (b) "Our test scores went from 62 to 98." (c) "If we shared the crayons equally, each table would get 24." (d) "On average, each day's rainfall was 3 millimeters away from the mean."
The mean of Rosa's 4 quiz scores is 85 points. Three of the scores are 80, 88 and 90. (a) What is the fourth score? (b) Find the range of the 4 scores. (c) Explain why knowing only the mean of 85 does not tell you how much Rosa's scores varied.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Measures of Center
Means and medians are correct, with units
One calculation error or missing units
Most centers wrong or missing
Measures of Variation
Ranges and MADs are correct, with units
One calculation error, or a negative distance used
Most variation measures wrong or missing
Center or Variation
Every number is correctly named as a center or a variation and explained in words
Most numbers named correctly
Center and variation confused
Explanations
Explains why one number cannot show both center and variation
Explanation is incomplete
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. All data sets are invented. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which of these is a measure of center?
Answer: C
The median is one number that stands for the whole data set as a typical value, so it is a measure of center. The range (choice A) and the mean absolute deviation (choice D) are measures of variation: they tell how much the values differ. The largest value (choice B) is a single value from the data and does not summarize all of them.
Question 2 of 20 · Multiple Choice
Invented data: the number of goldfish in 5 classroom tanks is 4, 6, 7, 9, 14. What is the mean?
Answer: B
Add the values and divide by how many there are: 4 + 6 + 7 + 9 + 14 = 40, and 40 ÷ 5 = 8 goldfish. Choice A is the median, the middle value. Choice C is the range, 14 - 4 = 10. Choice D is the total before dividing by 5.
Question 3 of 20 · Multiple Choice
Invented data: the lengths, in inches, of 6 sandwiches at a party are 8, 15, 6, 10, 6, 12. What is the range?
Answer: A
The range is the largest value minus the smallest value: 15 - 6 = 9 inches. Choice B subtracts the first value in the list from the last, 12 - 8, without finding the largest and smallest. Choice C is the mean, 57 ÷ 6 = 9.5. Choice D is the largest value.
Question 4 of 20 · Multiple Choice
Three friends have 5, 11 and 14 trading cards. If they share all the cards equally, how many does each friend get?
Answer: C
Sharing equally is finding the mean: 5 + 11 + 14 = 30 cards, and 30 ÷ 3 = 10 cards each. Choice B is the median, the middle value. Choice A is the range, 14 - 5 = 9. Choice D is the total before sharing.
Question 5 of 20 · Multiple Choice
Invented data: two coffee shops timed how many minutes 4 customers waited. Shop P: 3, 4, 4, 5. Shop Q: 1, 2, 5, 8. Both shops have a mean of 4 minutes. Which statement is true?
Answer: B
Shop Q's range is 8 - 1 = 7 minutes and Shop P's is 5 - 3 = 2 minutes, so Shop Q's times vary more. Choice C assumes that equal means give equal variation, but the mean is a measure of center only. Choice D confuses variation with center: the typical wait (the mean) is 4 minutes at both shops. Choice A reverses the ranges.
Question 6 of 20 · Multiple Choice
Invented data: the minutes Sam spent on each of 4 math problems are 3, 5, 8, 12. The mean is 7 minutes. What is the mean absolute deviation?
Answer: A
The distances from 7 are 4, 2, 1 and 5. They add to 12, and 12 ÷ 4 = 3 minutes. Choice D is the sum of the distances before dividing by 4. Choice C is the range, 12 - 3 = 9. Choice B is the mean itself, a measure of center.
Question 7 of 20 · Multiple Choice
The mean absolute deviation of a data set is 0. What must be true?
Answer: D
A MAD of 0 means every distance from the mean is 0, so every value equals the mean: there is no variation. The values can be any number, such as 7, 7, 7, so choice A is false. Choice B confuses the MAD with the values themselves. Choice C is false: a set such as 7, 7, 7 has three values and a MAD of 0, and a small set such as 3, 5 has a MAD of 1.
Question 8 of 20 · Multiple Choice
Invented data: 5 friends' allowances are $5, $6, $6, $7 and $11. The friend with $11 gets a raise to $21. What happens to the mean and the median?
Answer: C
The total goes from $35 to $45, so the mean goes from $7 to $9, up $2. The middle value is still $6, so the median does not change. The mean uses every value, but the median depends only on the middle. Choice B adds the $10 raise to the mean without dividing by 5. Choice A assumes the median moves too. Choice D swaps the two measures.
Question 9 of 20 · Multiple Choice
Which statement uses a measure of variation?
Answer: B
"Differ by up to 14 years" is a range, the largest age minus the smallest, which tells how much the ages vary. Choice A is a typical value, choice C is the mean as a fair share and choice D describes the median. All three are measures of center.
Question 10 of 20 · Multiple Choice
Invented data: a data set is 10, 12, 14, 16, 18, and its mean is 14. A new value is added to the set. Which new value would leave the mean at 14?
Answer: D
Adding a value equal to the mean keeps the fair share the same: the new total is 70 + 14 = 84, and 84 ÷ 6 = 14. Choice A makes the mean 80 ÷ 6, about 13.3, choice B makes it 88 ÷ 6, about 14.7, and choice C makes it 70 ÷ 6, about 11.7. Students who think 0 "adds nothing" forget that the number of values grows from 5 to 6.
Question 11 of 20 · Multiple Choice
Invented data: two classes took the same test. Both classes have a mean of 78 points. Class 1 has a MAD of 3 points, and Class 2 has a MAD of 11 points. What can you conclude?
Answer: A
The MAD is the average distance from the mean, so a MAD of 3 means Class 1's scores are, on average, only 3 points from 78. The means are equal, so neither class did better on average (choice B). The MAD does not count students (choice C). Choice D subtracts the MADs, 11 - 3 = 8, and treats variation as center.
Question 12 of 20 · Multiple Choice
A coach wants one number that shows how consistent a player's scores are from game to game. Which should she use?
Answer: D
Consistency is about how much the scores vary, so the coach needs a measure of variation such as the range. A small range means the scores stay close together. The mean (choice A) and the median (choice B) are measures of center: they tell a typical score, not how much the scores change. The total (choice C) grows with the number of games.
Question 13 of 20 · Multiple Choice
Invented data: the number of minutes 8 students spent on a science worksheet is 14, 18, 20, 21, 23, 25, 26, 29. What is the median?
Answer: B
There are 8 values, so the median is halfway between the 4th and 5th values: (21 + 23) ÷ 2 = 22 minutes. Choices A and C are only the 4th or the 5th value alone. Choice D is the range, 29 - 14 = 15 minutes.
Question 14 of 20 · Multiple Choice
The mean of 6 numbers is 15. What is the total of the 6 numbers?
Answer: C
The mean is the total divided by the number of values, so the total is 15 × 6 = 90. Choice A adds 15 + 6, choice B subtracts 15 - 6 and choice D divides 15 ÷ 6.
Question 15 of 20 · Short Answer
Invented data: the number of pull-ups 5 students did is 2, 4, 5, 8, 11. Find the mean and the mean absolute deviation. Say which number is the measure of center and which is the measure of variation.
Total: 2 + 4 + 5 + 8 + 11 = 30, so the mean is 30 ÷ 5 = 6 pull-ups. Distances from 6: 4, 2, 1, 2, 5, which add to 14, so the MAD is 14 ÷ 5 = 2.8 pull-ups. The mean is the measure of center (a typical number of pull-ups). The MAD is the measure of variation (on average, a student is 2.8 pull-ups away from 6).
Question 16 of 20 · Short Answer
Write two data sets of 4 values that both have a mean of 10, where one set has a range of 0 and the other has a range of 8.
Many answers work. One pair: 10, 10, 10, 10 (total 40, mean 10, range 10 - 10 = 0) and 6, 9, 11, 14 (total 40, mean 10, range 14 - 6 = 8). Any answer works if each set of 4 values has a total of 40, the first set has four equal values and the second set's largest value minus its smallest value is 8.
Question 17 of 20 · Short Answer
A student says, "The mean tells you everything about a data set." Use an example to explain why this is not true.
The mean is one number for the center, so it cannot show how much the values vary. Example: 5, 5, 5 and 1, 4, 10 both have a mean of 15 ÷ 3 = 5, but the first set has a range of 0 and the second has a range of 10 - 1 = 9. A measure of variation is needed to show the difference.
Question 18 of 20 · Short Answer
Invented data: the prices of 4 paperback books are $6, $8, $9 and $13. Find the median and the range, and explain in words what each number tells.
In order, the two middle prices are $8 and $9, so the median is (8 + 9) ÷ 2 = $8.50: half the books cost less than $8.50 and half cost more. The range is 13 - 6 = $7: the most expensive book costs $7 more than the cheapest. The median is a measure of center; the range is a measure of variation.
Question 19 of 20 · Short Answer
Invented data: a plant's height grew 2, 3, 3, 4 centimeters in 4 weeks. Its mean weekly growth is 3 centimeters. Find the MAD, and explain what it means.
Distances from 3: 1, 0, 0, 1, which add to 2, so the MAD is 2 ÷ 4 = 0.5 centimeter. On average, one week's growth is only half a centimeter away from the mean growth of 3 centimeters, so the plant grew at a very steady rate.
Question 20 of 20 · Short Answer
Explain the difference between a measure of center and a measure of variation. Give one example of each.
A measure of center is one number that summarizes all the values with a typical value, for example the mean (the fair share) or the median (the middle value). A measure of variation is one number that tells how much the values differ from each other, for example the range (largest minus smallest) or the MAD (the average distance from the mean).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 6.SP.A.3 mean?
6.SP.A.3 means students recognize two jobs that a single number can do for a data set. A measure of center, like the mean or the median, gives one typical value that stands for all the values. A measure of variation, like the range or the mean absolute deviation, gives one number that tells how much the values differ.
Is 6.SP.A.3 about calculating the mean?
Partly: students do find means and medians, but the standard is mainly about understanding what those numbers do. The key idea is that a mean summarizes all the values with one number and a range or MAD describes the variation with one number. Using these measures to summarize real data sets in full is the focus of 6.SP.B.5.
What is the difference between the mean and the median?
The mean is the fair share, the total divided by the number of values, so every value affects it. The median is the middle value in order, so it depends only on the middle of the data. If one value becomes much larger, the mean goes up, but the median can stay the same.
What is a measure of variation?
A measure of variation is one number that tells how much the values in a data set differ from each other. In grade 6, the main ones are the range (largest minus smallest), the interquartile range (the range of the middle half of the data, taught in 6.SP.B.5) and the mean absolute deviation. A bigger number means the values are more spread out, and 0 means every value is the same.
How do you find the mean absolute deviation?
Find the mean, then find how far each value is from the mean, then find the mean of those distances. For 4, 6, 11: the mean is 21 ÷ 3 = 7, the distances are 3, 1 and 4, and the MAD is 8 ÷ 3, about 2.7. Distances are never negative.
Why do we need both a measure of center and a measure of variation?
Because each one answers a different question. Two groups can have the same typical value but very different amounts of variation, such as a steady player who scores 12 points every game and a streaky player who averages 12 but scores anywhere from 2 to 25. Reporting both numbers tells the real story.
What grade learns 6.SP.A.3?
It is a grade 6 standard in the Statistics and Probability domain. In grade 7, students use measures of center and variation to compare two groups (7.SP.B.3 and 7.SP.B.4). In high school statistics, students add the standard deviation, another measure of variation based on distances from the mean (HSS.ID.A.2).
What mistakes do students often make with the mean and the range?
A common mistake is dividing the total by the wrong number, such as the largest value instead of the number of values. Another is forgetting to put the values in order before finding the median or the range. Many students also think a large mean means a large variation, but center and variation are separate ideas.
Do measures of center and variation have units?
Yes: both use the same units as the data. If the data are minutes, the mean, the median, the range and the MAD are all in minutes. Asking students to write the unit every time also helps them say what the number means, such as "a typical bus ride is 15 minutes" or "the rides differ by up to 9 minutes".
How can parents practice measures of center and variation at home?
Use everyday numbers, such as the minutes it takes to get ready each morning for a week. Ask your child to find a typical time (the mean or the median) and how much the times change (the range). Then ask which number would help plan the morning better.
07
Related Standards
6 standards
These standards connect to 6.SP.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.MD.B.2Prerequisite
Make line plots of measurement data and share a total equally, as in beakers of liquid
Lesson coming soon
6.SP.A.1Prerequisite
Recognize a statistical question as one that expects variability in the data
