ACT Math study guide · Chapter 31
Center and Spread
The mean is the sum of all values divided by the count. The median is the middle value once the data is ordered, or the average of the two middle values if the count is even. The mode is the value that appears most frequently. A missing-value problem gives the mean and all but one value; multiplying the mean by the count recovers the total, and subtracting the known values isolates the missing one.
Spread describes how far the data is scattered, not where it's centered. The range is the maximum minus the minimum. The interquartile range, IQR, is Q3 minus Q1, the middle 50% of the data. Standard deviation measures the typical distance of each value from the mean; a data set with values spread farther from the mean has a larger standard deviation, even if two data sets share the same mean.
Where this appears as a model: several groups combined into one average. Two class sections of 20 and 15 students with means of 78 and 85 combine to an overall mean of 81, found by weighting each mean by its class size instead of averaging 78 and 85 directly.
▸ Missing-value problems
Worked Example Four numbers have a mean of 15. Three of them are 9, 14, and 21. Find the fourth.
Recover the total by multiplying the mean by the count, then subtract the known values.
Total = 15 × 4 = 60
60 − (9 + 14 + 21) = 16
▸ Comparing spread across data sets
Two data sets can share the same mean while having very different spreads. A data set whose values sit far from the mean has a larger standard deviation than one whose values cluster tightly around it, regardless of what that shared mean actually is.
▸ Distribution shape and histograms
A distribution has shape as well as center and spread. A roughly symmetric distribution behaves similarly on both sides of its center. A right-skewed distribution has a longer tail toward larger values; those large values pull the mean to the right, so the mean is usually greater than the median. A left-skewed distribution has a longer tail toward smaller values, pulling the mean to the left, so the mean is usually less than the median.
A histogram displays how often values fall into each interval, so read its overall shape before calculating anything. Where is the center, how wide is the spread, is it symmetric or skewed, and are there isolated gaps or unusually extreme groups? When strong skew or outliers are present, the median and IQR are usually more resistant summaries of the data than the mean and standard deviation, which are both pulled toward extreme values.
Worked Example A data set has mean 62 and median 51. What does this suggest about the distribution's shape?
Compare the mean and median directly.
The mean (62) is greater than the median (51)
This pattern indicates a right-skewed distribution, with a tail of unusually high values pulling the mean upward
▸ Practice
Question 1
- AA. Left-skewed
- BB. Right-skewed
- CC. Roughly symmetric
- DD. Cannot be determined
Question 2
- FF. Right-skewed
- GG. Roughly symmetric
- HH. Left-skewed
- JJ. Uniform
Question 3
- AA. The mean
- BB. The median
- CC. Both are equally resistant
- DD. Neither is resistant
Question 4
- FF. The mean and median are likely close to each other
- GG. The distribution must be skewed
- HH. The standard deviation cannot be calculated
- JJ. The median is undefined
▸ Answers and Explanations
Show the answers and explanations
1: B. The mean (45) is greater than the median (38); a tail of high values pulling the mean upward is the signature of a right-skewed distribution. Choice A describes the opposite pattern, where the mean would be less than the median.
2: H. A long tail toward small values pulls the mean toward the lower end while most data sits higher up; this is the definition of a left-skewed distribution. Choice F describes the opposite tail direction.
3: B. The median depends only on the middle value's position, not on how far extreme values are from it, so it resists the pull of outliers. The mean shifts directly toward extreme values. Choice A selects the measure that is actually most affected by the skew described.
4: F. A roughly symmetric, single-peaked distribution with no unusual gaps or extreme groups typically has its mean and median close together, since neither is being pulled by a long tail. Choice G contradicts the symmetric shape stated in the question.
▸ Practice
Question 1
- AA. 11
- BB. 13
- CC. 10
- DD. 12.5
Question 2
- FF. 3
- GG. 9
- HH. 7
- JJ. 5.5
Question 3
- AA. 14 and 22.4
- BB. 14 and 14
- CC. 22.4 and 14
- DD. 14 and 60
Question 4
- FF. 55
- GG. 15
- HH. 20
- JJ. 35
Question 5
- AA. 6
- BB. 13
- CC. 8
- DD. 9
Question 6
- FF. Set B
- GG. Set A
- HH. They are equal
- JJ. Cannot be determined
Question 7
- AA. 17
- BB. 15
- CC. 19
- DD. 13
Question 8
- FF. 15
- GG. 19
- HH. 11
- JJ. 4
Question 9
- AA. 8 and 8
- BB. 16 and 16
- CC. 16 and 8
- DD. 8 and 16
Question 10
- FF. 18
- GG. 13
- HH. 31
- JJ. 49
Question 11
- AA. 23
- BB. 27
- CC. 9
- DD. 31
Question 12
- FF. 7
- GG. 4
- HH. 9
- JJ. 8
Question 13
- AA. Set Y
- BB. Set X
- CC. They are equal
- DD. Cannot be determined
Question 14
- FF. 17
- GG. 25
- HH. 29
- JJ. 21
Question 15
- AA. 16
- BB. 18
- CC. 20
- DD. 14
Question 16
- FF. 15
- GG. 27
- HH. 21
- JJ. 33
Question 17
- AA. 6
- BB. 11
- CC. 9
- DD. 14
Question 18
- FF. 16
- GG. 13
- HH. 29
- JJ. 45
Question 19
- AA. 33
- BB. 5
- CC. 22
- DD. 28
Question 20
- FF. Set A
- GG. They are equal
- HH. Cannot be determined
- JJ. Set B
▸ Answers and Explanations
Show the answers and explanations
1: D. (8+10+10+12+15+20) ÷ 6 = 75 ÷ 6 = 12.5. Choice A comes from an arithmetic slip while summing the values before dividing.
2: F. 3 appears three times, more often than any other value. Choice G reports the largest value in the set instead of the one that appears most frequently.
3: A. Original mean = 70/5 = 14; with 60 replacing 18, mean = 112/5 = 22.4. Choice B reports the original mean twice, forgetting to recompute after replacing 18 with 60.
4: G. IQR = Q3 − Q1 = 35 − 20 = 15. Choice F adds Q1 and Q3 instead of subtracting Q1 from Q3.
5: C. Range = maximum − minimum = 13 − 5 = 8. Choice A comes from an arithmetic slip while subtracting the minimum from the maximum.
6: F. A wider spread of values away from the mean produces a larger standard deviation, regardless of the shared mean. Choice G reverses which data set has the wider spread.
7: A. (8+13+17+22+25)÷5=85÷5=17. Choice B comes from an arithmetic slip while summing the values before dividing.
8: F. Ordered: 4,11,15,19,26. The middle value is 15. Choice G reports a value adjacent to the true median instead of the middle value itself.
9: C. New total: 6+7+8+9+50=80; new mean: 80÷5=16. The median stays at the middle of the ordered list 6,7,8,9,50, which is 8. Choice D swaps the new mean and the median, reporting them in the wrong order.
10: G. IQR=Q3−Q1=31−18=13. Choice F reports Q1 itself instead of subtracting it from Q3.
11: B. Range=maximum−minimum=31−4=27. Choice A comes from an arithmetic slip while subtracting the minimum from the maximum.
12: H. 9 appears three times, more often than any other value. Choice G reports a value that appears only once instead of the one appearing most frequently.
13: A. A wider range of values away from the shared mean means Set Y has the larger standard deviation. Choice B reverses which data set has the wider spread.
14: J. Total=21×5=105; 105−(13+19+24+28)=21. Choice F comes from an arithmetic slip while recovering the total or subtracting the known values.
15: B. (9+14+18+23+26)÷5=90÷5=18. Choice A comes from an arithmetic slip while summing the values before dividing.
16: H. Ordered: 8,15,21,27,33. The middle value is 21. Choice G reports a value adjacent to the true median instead of the middle value itself.
17: B. 11 appears three times, more often than any other value. Choice A reports the smallest value in the set instead of the one that appears most frequently.
18: G. IQR=Q3−Q1=29−16=13. Choice F reports Q1 itself instead of subtracting it from Q3.
19: D. Range=maximum−minimum=33−5=28. Choice A reports the maximum value itself instead of subtracting the minimum from it.
20: J. A wider range of values away from the shared mean means Set B has the larger standard deviation. Choice F reverses which data set has the wider spread.