Why it matters on the HSPT
Data questions are short and the arithmetic is light, which makes them some of the best points on the Mathematics section. The test asks for the mean, median, mode, or range of a short list, asks you to read a number off a table or bar graph and do one step with it, and asks a simple probability from a bag of marbles or a spinner. The one twist that catches students is the average worked backward, where you are given the target average and asked for a missing value.
The rules
Mean. Add the values and divide by the count. The mean of 4, 8, and 9 is 21 divided by 3, which is 7. The mean does not have to be one of the values.
Median. Put the values in order and take the middle one. For 3, 9, 4, 7, 6, the order is 3, 4, 6, 7, 9, and the median is 6. With an even count, average the two middle values: for 2, 5, 8, 11, the median is the average of 5 and 8, which is 6.5. Always sort first; a median read from an unsorted list is the test's favorite wrong answer.
Mode. The value that appears most often. A list can have one mode, more than one, or none if nothing repeats.
Range. The largest value minus the smallest. The range of 3, 9, 4, 7, 6 is 9 minus 3, which is 6.
Working the mean backward. The total of the values is the mean times the count. If you know the mean and all but one value, find the total, then subtract what you have. This is the same move as the reversed percent problems: from the average to the whole, then to the missing piece.
Tables and graphs. Read the axis labels and the units before the numbers. A bar graph's height is read against the scale on the side, and the scale does not always count by ones. Questions usually ask for a difference between two bars, a sum of several, or which category is largest or smallest. Do the reading on the graph with your pencil, marking the values you use.
Probability. The probability of an event is the number of favorable outcomes divided by the total number of equally likely outcomes, written as a fraction in lowest terms. A bag with 4 red, 3 blue, and 1 green marble gives a probability of 3 over 8 for blue. The probability of something impossible is 0; of something certain, 1. The probability that an event does not happen is 1 minus the probability that it does. Two dice or two coins are the only multi-step cases, and the test handles them by counting outcomes: a coin flipped twice has four outcomes, and exactly one of them is two heads.
Worked examples
Example 1
What is the mean of 12, 15, 9, and 16?
- A.12
- B.13
- C.13.5
- D.15
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Answer: B: 13
Add: 12 plus 15 plus 9 plus 16 is 52. Divide by 4: 13. Choice C, 13.5, is the median of the sorted list, 9, 12, 15, 16, a different measure.
Example 2
What is the median of 7, 2, 9, 4, 11?
- A.4
- B.6.6
- C.7
- D.9
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Answer: C: 7
Sort first: 2, 4, 7, 9, 11. The middle value is 7. Choice D, 9, is the middle of the unsorted list. Choice B, 6.6, is the mean.
Example 3
For the data set 5, 8, 8, 10, 19, which measure is the largest?
- A.mean
- B.median
- C.mode
- D.range
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Answer: D: range
Compute all four. The mean is 50 divided by 5, which is 10. The median of the sorted list is 8. The mode is 8. The range is 19 minus 5, which is 14. The largest is the range. A single large value like 19 pulls the mean and the range up but leaves the median and mode alone.
Example 4
Ana scored 84, 90, and 87 on three tests. What must she score on the fourth to have a mean of 90?
- A.90
- B.93
- C.96
- D.99
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Answer: D: 99
A mean of 90 over four tests needs a total of 360. So far she has 84 plus 90 plus 87, which is 261. She needs 360 minus 261, which is 99. Choice A is the trap for thinking the next score just has to equal the target average.
Example 5
A bar graph shows books read this month: Mia 12, Leo 8, Sam 20, Ivy 9. How many more books did Sam read than Leo and Ivy combined?
- A.3
- B.8
- C.11
- D.12
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Answer: A: 3
Leo and Ivy combined read 8 plus 9, which is 17. Sam read 20, so the difference is 20 minus 17, which is 3. Choice D, 12, is Sam minus Leo alone, the trap for reading only one of the two bars the question names.
Example 6
A bag holds 5 red, 3 green, and 2 yellow marbles. What is the probability of drawing a green marble?
- A.3/7
- B.3/10
- C.1/3
- D.3/5
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Answer: B: 3/10
Total marbles are 5 plus 3 plus 2, which is 10. Three are green, so the probability is 3 over 10. Choice A divides by the non-green marbles, and choice C divides by the number of colors.
Example 7
A spinner has 8 equal sections numbered 1 to 8. What is the probability of spinning a number greater than 6?
- A.1/8
- B.1/4
- C.3/8
- D.1/2
Show answer
Answer: B: 1/4
Greater than 6 means 7 or 8, which is 2 sections out of 8. Two over 8 simplifies to 1 over 4. Choice C counts 6, 7, and 8, but 6 is not greater than 6.
Common mistakes
- Reading the median from an unsorted list. Sort first, every time.
- Confusing the mean with the median. The mean uses every value; the median only cares about position.
- Treating the target average as the needed score. Find the total first, then subtract.
- Counting the wrong total in probability. The denominator is every outcome, including the favorable ones.
- Misreading a graph's scale. Check what each gridline counts by before reading a bar.
Practice
These skills appear in word problems about scores and surveys, in arithmetic items on averages, and in the geometric comparison items that use bar graphs. The fractions lesson covers simplifying the probabilities.