Statistical Analysis
Statistical Analysis
Statistical analysis sounds like dental surgery. Scientific and sticky and gross. But SAT statistical analysis is actually not so bad. On these questions, the SAT gives you a data set—a collection of measurements or quantities. An example of a data set is the set of math test scores for the 20 students in Ms. Mathew’s fourth-grade class:
71, 83, 57, 66, 95, 96, 68, 71, 84, 85, 87, 90, 88, 90, 84, 90, 90, 93, 97, 99
You are then asked to find one or more of the following values:
  1. Arithmetic Mean
  2. Median
  3. Mode
  4. Range
Arithmetic Mean (a.k.a. Average)
Arithmetic mean means the same thing as average. It’s also the most commonly tested concept of statistical analysis on the SAT. The basic rule of finding an average isn’t complicated: It’s the value of the sum of the elements contained in a data set divided by the number of elements in the set.
Take another look at the test scores of the 20 students in Ms. Mathew’s class. We’ve sorted the scores in her class from lowest to highest:
57, 66, 68, 71, 71, 83, 84, 84, 85, 87, 88, 90, 90, 90, 90, 93, 95, 96, 97, 99
To find the arithmetic mean of this data set, sum the scores and then divide by 20, since there are 20 students in her class:
But the SAT is Sneaky When It’s Mean
But that’s not the way that the SAT usually tests mean. It likes to be more complicated and conniving. For example,
If the average of four numbers is 22, and three of the numbers are 7, 11, and 18, then what is the fourth number?
Here’s the key: If you know the average of a group and also know how many numbers are in the group, you can calculate the sum of the numbers in the group. The question above tells you that the average of the numbers is 22 and that there are four numbers in the group. If the average of four numbers is 22, then the four numbers, when added together, must equal 4 × 22 = 88. Since you know three of the four numbers in the set, and since you now know the total value of the set, you can write
Solving for the unknown number is easy. All you have to do is subtract the sum of 7, 11, and 18 from 88: x = 88 – (7 + 11 + 18) = 88 – 36 = 52.
As long as you realize that you can use an average to find the sum of all the values in a set, you can solve pretty much every question about arithmetic mean on the SAT:
The average of a set of seven numbers is 54. The average of three of those seven numbers is 38. What is the average of the other four numbers?
This question seems really tough, since it keeps splitting apart its set of seven mysterious numbers. Students often freak out when SAT questions ask them for numbers that seem impossible to determine. Chill out. You don’t have to know the exact numbers in the set to answer this problem. All you have to know is how averages work.
There are seven numbers in the entire set, and the average of those numbers is 54. The sum of the seven numbers in the set is 7 × 54 = 378. And, as the problem states, three particular numbers from the set have an average of 38. Since the sum of three items is equal to the average of those three numbers multiplied by three, the sum of the three numbers in the problem is 3 × 38 = 114. Once you’ve got that, you can calculate the sum of the four remaining numbers, since that value must be the total sum of the seven numbers minus the sum of the mini-set of three: 378 – 114 = 264. Now, since you know the total sum of the four numbers, you can get the average by dividing by 4: 264 ÷ 4 = 66.
And here’s yet another type of question the SAT likes to ask about mean: the dreaded “changing mean” question.
The mean age of the 14 members of a scuba diving club is 34. When a new member joins, the mean age increased to 37. How old is the new member?
Actually, you shouldn’t dread “changing mean” questions at all. They’re as simple as other mean questions. Watch. Here’s what you know from the question: the original number of members, 14, and the original average age, 34. And you can use this information to calculate the sum of the ages of the members of the original group by multiplying 14 × 34 = 476. From the question, you also know the total members of the group after the new member joined, 14 + 1 = 15, and you know the new average age of the group, 37. So, you can find the sum of the ages of the new group as well: 15 × 37 = 525. The age of the new member is just the sum of the age of the new group minus the sum of the age of the old group: 555 – 476 = 79. That is one ancient scuba diver.
Median
The median is the number whose value is exactly in the middle of all the numbers in a particular set. Take the set {6, 19, 3, 11, 7}. If the numbers are arranged in order of value, you get
{3, 6, 7, 11, 19}
It’s clear that the middle number in this group is 7, so 7 is the median.
If a set has an even number of items, it’s impossible to isolate a single number as the median. Here’s the last set, but with one more number added:
{3, 6, 7, 11, 19, 20}
In this case, the median equals the average of the two middle numbers. The two middle numbers in this set are 7 and 11, so the median of the set is (7+11) /2 = 9.
Mode
The mode is the number within a set that appears most frequently. In the set {10, 11, 13, 11, 20}, the mode is 11, since it appears twice and all the others appear once. In a set where more than one number appears at the same highest frequency, there can be more than one mode: The set {2, 2, 3, 4, 4} has modes of 2 and 4. In the set {1, 2, 3, 4, 5}, where all of the numbers appear an equal number of times, there is no mode.
Range
The range measures the spread of a data set, or the difference between the smallest element and the largest. For the set of test scores in Ms. Mathew’s class, {57, 66, 68, 71, 71, 83, 84, 84, 85, 87, 88, 90, 90, 90, 90, 93, 95, 96, 97, 99}, the range is 99 – 57 = 42.
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