6th Grade · 6.SP.A.2

The shape of data in 6th grade

The shape of data is the pattern the values make when you plot them, such as one hill in the middle or a pile on one side. In 6th grade, your child learns that every set of data has a distribution that can be described by its center, its spread and its shape. Your child describes data in these words before learning to calculate measures such as the median.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

What it means, in plain words

A distribution is the whole set of values collected to answer a question and how often each value appears. A dot plot shows it as stacks of dots above a number line, with one dot for each value. A histogram shows it as bars, where each bar counts the values in a range, such as 10 to 19.

Your child describes a distribution in three ways. The center is a typical value, near the middle of the data. The spread is how far the values range, from the smallest to the largest. Center and spread together already say a lot. Two teams can have the same typical score while one team's scores are much more spread out.

The shape of data describes the overall pattern. A symmetric shape looks about the same on both sides of the middle, like a hill. A skewed shape piles up on one side and has a long tail on the other side. Your child also looks for peaks (the tallest stacks), clusters (groups of values close together), gaps (empty stretches) and values far from the rest. The shape of data in 6th grade is described in words. Ways to calculate the center and the spread come later in the year.

Your child should be able to:

Read a dot plot or a histogram and say what one dot or one bar stands for
Give a typical value as the center of a data set
Describe the spread from the smallest value to the largest
Describe the shape as symmetric or skewed and point out peaks, clusters and gaps

Example

A soccer team scored 1, 2, 2, 3, 3, 3, 4, 4, 5 and 9 goals in its 10 games. Describe the center, the spread and the shape of the data.

Make a dot plot with one dot for each game, stacked above the number of goals.
The tallest stack is at 3, and the two middle values of the list are also 3. A typical game had about 3 goals, so the center is about 3.
The values run from 1 goal to 9 goals. That is the spread.
Most games pile up between 1 and 5 goals, with a peak at 3. One game, with 9 goals, sits far to the right after a gap.

Example

Two basketball players scored these points in 9 games. Player A: 4, 5, 5, 6, 6, 6, 7, 7, 8. Player B: 1, 2, 4, 6, 6, 6, 8, 10, 11. How are the two distributions alike, and how are they different?

Both data sets are symmetric around 6, so both centers are 6 points.
Player A's scores run from 4 to 8 points, a spread of $8 - 4 = 4$ points.
Player B's scores run from 1 to 11 points, a spread of $11 - 1 = 10$ points.
The centers match, but Player B's scores are much more spread out from game to game.

Where children usually get stuck

Looking at a histogram, your child says the typical value is 12 because the tallest bar reaches 12.
The child reads the height of the bar, which counts how many values are in that range. The values themselves are the labels along the bottom.
What helps. Put a finger at the bottom of the tallest bar and read the label there. That label, not the height of the bar, is near the center.
For the soccer team, your child says the typical number of goals is 5, because 5 is halfway between 1 and 9.
The child takes the middle of the spread instead of the middle of the data. The one game with 9 goals pulls that halfway point up, while 7 of the 10 games had 2, 3 or 4 goals.
What helps. Cover the 9 with your hand and ask where most of the dots are. The center belongs where the data piles up.
Your child calls a dot plot symmetric because it has one peak, even though the dots trail off far to one side.
One clear peak looks tidy, so the child calls the shape symmetric. Symmetric means that both sides of the peak look about the same.
What helps. Imagine folding the plot in half at the peak. If the two sides do not match, the shape is skewed, with a tail on the longer side.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The dot plot below shows how many books students read last month.
Books read
Select the interval from the smallest plotted value to the largest plotted value.
A
2-8
B
3-7
C
3-8
D
4-8
Show the answer and how to solve it
The answer is 3-8 because 3 is the smallest plotted value and 8 is the largest plotted value.
1. 1Look at the leftmost dot on the plot. It is above 3, so the smallest value is 3.
2. 2Now look at the rightmost dots. They are above 8, so the largest value is 8.
3. 3The interval from the smallest value to the largest value is 3-8, which is choice-c.
Common mistake. A common mistake is choosing 2-8 because the axis shows a tick at 2. There is no dot at 2, so the interval should start at 3.
The idea underneath. To find an overall spread interval on a dot plot, use the two extreme data values — the leftmost and rightmost dots — not the middle cluster.
2On level
The histogram below shows quiz scores for a class.
Jordan's claim: "The spread interval is 10-19."
Select the best explanation of Jordan's mistake.
A
The student included an outside interval with no data.
B
The student kept the right end but forgot the first interval on the left.
C
The student kept the left end but stopped at the tallest bar.
D
The student used only the middle busy section instead of the complete range of the data.
Show the answer and how to solve it
The best explanation is that Jordan used only the middle busy section instead of the complete range of the data. The full range must include all nonempty intervals, from 5-9 through 20-24.
1. 1Look at the histogram and find every interval that has a bar above 0.
2. 2The first interval with data is 5-9 and the last interval with data is 20-24. So the full spread interval is 5-24.
3. 3Jordan's claim, 10-19, covers only the middle intervals 10-14 and 15-19. It leaves out the nonempty bars at 5-9 on the left and 20-24 on the right.
4. 4The best explanation is that Jordan used only the middle busy section instead of the complete range of the data.
Common mistake. A common mistake is calling 10-19 the spread because most scores fall in that middle cluster. Spread must still include the nonempty end intervals 5-9 and 20-24.
The idea underneath. In a histogram, spread is read from the first nonempty bin to the last nonempty bin, not from the tallest or most crowded part of the graph.
3On level
Make the completed dot plot roughly symmetric.
Add exactly one point.
Tap or click above a number to place marks there. Tap existing marks to remove them.
Hours of Sleep
Hours of sleep
Show the answer and how to solve it
Add the point at 6. Then the stack heights are 1, 3, 4, 3, 1, so the completed dot plot is roughly symmetric.
1. 1For symmetry, use 7 as the center because it is halfway between the end values, 5 and 9. Compare stack heights at values the same distance from 7.
2. 2Compare the values the same distance from 7.
3. 3The stack at 6 is missing 1 point to match the stack at 8, so the new point should go at 6.
4. 4Now the plot has matching heights on both sides of 7.
Common mistake. A common mistake is adding the point at 7 or 8 because those stacks already look large. For symmetry, you need matching heights on opposite sides of 7, so the missing point belongs at 6.
The idea underneath. To make a dot plot roughly symmetric, compare stack heights at values the same distance from the center and make those pairs match.
4Stretch
The histogram below shows students' homework times in minutes. Priya's claim: "54 minutes is a good typical value for this data."
Select the statement that best explains whether Priya's claim is a good estimate of the center.
A
No, because a center should be near the middle of where most of the data are, and the middle of this histogram is in the 60-70 interval, which is above 54.
B
Yes, because 54 is close to 55, which is halfway between the ends of the data range shown.
C
Yes, because 54 falls inside the 50-60 interval, so it can be the center.
D
No, because the center should be the tallest interval, which is 60-70.
Show the answer and how to solve it
54 minutes is not a good typical value because the middle of the data falls in the 60-70 interval, which is above 54.
1. 1The tallest bar is 60-70 with 11 students, and the next bar, 70-80, has 8 students. The main group of data is well to the right of 54.
2. 2Count students below 60, then how many the count reaches by the end of 60-70.
3. 3There are 33 students total, so the middle is the 17th value. Only 12 students are below 60, and the count reaches 23 inside 60-70, so the 17th value falls in that interval. That puts the middle of the data somewhere in 60-70. The midpoint of that interval, 65, is only a rough estimate for it.
4. 454 is below the whole 60-70 interval, so it is too low to be the middle of the data. The correct statement is the one saying the middle of this histogram is in the 60-70 interval, which is above 54.
Common mistake. A common mistake is accepting 54 because it is close to 55, the midpoint of the 20-90 range. The center should come from where most of the data are, not from the endpoints of the displayed range.
The idea underneath. To judge a typical value on a histogram, look for the middle of the main group of bars, not the midpoint of the range or a single tallest bar.