When preparing for the SAT or ACT, students will eventually encounter questions on unfamiliar topics from probability and statistics. For example, students typically have not learned how to calculate the expected value of a random discrete variable, which we have explored previously. Standard deviation is another commonly tested statistics concept that often challenges students taking the test, regardless of what level of math they are studying in school. Fortunately, you don’t need to memorize the formula for standard deviation! Instead, a basic understanding of what standard deviation represents and some practice comparing datasets will give you the tools you need to answer these questions correctly.
What Is Standard Deviation?
In simple terms, standard deviation is a measure of how “spread out” the values in a dataset are. If the numbers in a dataset are clustered close to the mean (average), the standard deviation is small. If the numbers are more spread out, the standard deviation is larger. Think of it as a way to quantify consistency: the smaller the standard deviation, the more consistent the data values are.
Key Points About Standard Deviation
The Role of the Mean: Standard deviation revolves around the mean of the dataset. It measures how far the data points tend to deviate from this central value. However, for test purposes, you rarely need to calculate the mean or the deviations yourself.
Range as a Clue: On the SAT and ACT, standard deviation questions often focus on comparing two datasets. For example, you might see two sets of test scores or two distributions of temperatures. When the number of data points and the overall distribution shapes are similar, the dataset with the larger range (the difference between the highest and lowest values) usually has the larger standard deviation. This makes sense because a larger range generally indicates more variability.
Symmetrical Distributions: Many SAT and ACT questions assume symmetrical distributions (e.g., bell curves or evenly spaced values). In these cases, it’s usually safe to correlate range with standard deviation, as they typically increase together.
Example Question 1
Dataset A: 10, 12, 14, 16, 18
Dataset B: 8, 10, 14, 18, 20
Which dataset has the larger standard deviation?
Both datasets have five values with equal mean, and their distributions are symmetrical. Dataset A’s range is 18 – 10 = 8, while Dataset B’s range is 20 – 8 = 12. In Dataset B, the 10 and 18 are farther from the mean of 14 than the 12 and 16 from Dataset A, and the same goes for the 8 and 20 from Dataset B compared to the 10 and 18 from Dataset A. The values in Dataset B are more spread out, so Dataset B has the larger standard deviation.
Example Question 2
The following chart shows the frequency of test scores for two different classes, Class A and Class B. Each score represents the number of points earned out of 10.
| Frequency Distribution of Test Scores | ||||||||||
| Score | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Class A | 0 | 1 | 2 | 3 | 5 | 4 | 3 | 2 | 1 | 0 |
| Class B | 0 | 0 | 1 | 4 | 10 | 4 | 2 | 0 | 0 | 0 |
Which class’s test scores have the greater standard deviation, and why?
Correct Answer: Class A
Standard deviation measures the spread of data around the mean. In this case, Class A’s scores are distributed across a wider range of values (from 2 to 9), with frequencies gradually increasing and then decreasing. This spread shows a greater variation in scores. Class B’s scores, on the other hand, are tightly clustered around the middle (most students scored 5), with no data outside the range of 3 to 7. Be careful: even though both datasets have the same range of possible scores (1–10), Class A’s actual distribution of scores is much more spread out, which results in a higher standard deviation.
Tips for Answering Standard Deviation Questions on the SAT and ACT
Make Use of the Range: If you’re comparing datasets with the same number of elements and symmetrical distributions, the range is probably your best clue. A greater range usually indicates that the corresponding dataset’s standard deviation is also greater.
Look for Symmetry: Most test questions won’t include datasets with wildly irregular or skewed distributions. If you notice symmetry in the values, the relationship between range and standard deviation should hold true.
Don’t Overthink: You don’t need to calculate anything. Focus on the conceptual meaning of standard deviation and rely on visual patterns or simple comparisons.
Why This Approach Works
The SAT and ACT don’t expect students to perform calculations for standard deviation. Instead, the tests assess your ability to interpret and compare data. With a basic definition for standard deviation and by understanding the relationship between range and standard deviation, you can confidently tackle these questions without getting bogged down in formulas.
Final Recommendation
You don’t have to be in AP Statistics to correctly answer questions about standard deviation! When comparing datasets on the SAT or ACT, remember the definition we brought up earlier: if the datasets have a similar number of elements and symmetrical distributions, the dataset with values that deviate farther from the mean has the greater standard deviation. Often, the dataset with the greater range will also have a greater standard deviation. Keep these principles in mind, and you’ll be well-prepared to handle any standard deviation question that comes your way!
