Mean, median and spread — Digital SAT Math practice
Measures of center and spread; effect of outliers.
Key terms
- Mean: The arithmetic average calculated by dividing the sum of all data points by the total number of points.
- Median: The middle value in an ordered dataset; if even, it is the average of the two central values.
- Outlier: An extreme value far from the rest of the data that heavily skews the mean but barely affects the median.
- Range: The simplest measure of spread found by subtracting the minimum value from the maximum value in the set.
- Standard Deviation: A number representing the typical distance of each data point from the mean; larger values indicate more spread.
- Mode: The value that appears most frequently in a dataset; a set may have none or multiple modes.
Common traps
- Assuming the median equals the mean without checking symmetry, since outliers can shift the mean away from the median by an unpredictable amount.
- Calculating the median by averaging all values instead of ordering them and finding the center position first.
- Believing that a larger range guarantees a larger standard deviation, since range ignores how data clusters around the mean.
Key ideas to know
Finding a Missing Test Score
The mean of five test scores is 82. Four scores are 78, 85, 90, and 76. What is the fifth score?
Multiply the mean by the count to get the total sum (82 × 5 = 410). Subtract the known scores from the total sum (410 − 78 − 85 − 90 − 76 = 81). The missing score is 81.
Watch out: Dividing the sum of known values by their count instead of using the full dataset size to find the total sum first.
Calculating the Median
Find the median of the data set: 14, 9, 21, 17, 5, 12.
Order the values from least to greatest: 5, 9, 12, 14, 17, 21. Since there is an even number of values, average the two middle numbers: (12 + 14) / 2 = 13.
Watch out: Forgetting to order the list first or averaging the wrong pair when the count is even.
Outliers and the mean
A student believes that adding a large outlier to a data set will change the median more than it changes the mean. What is wrong with this belief?
The belief is backwards. An outlier pulls the mean much more than the median, because the mean uses every value while the median depends only on position.
Watch out: Assuming a measure of center always shifts a lot with extreme values. The median resists outliers.
Comparing Spread with Range and SD
Data set A has a range of 50 and SD of 12. Data set B has a range of 40 and SD of 18. Which is more spread out, and why?
Data set B is more spread out because standard deviation measures the typical distance from the mean. A larger SD indicates greater overall variability, even if its range is smaller.
Watch out: Relying solely on range to judge spread, which only looks at two values and ignores how all data points are distributed around the mean.
Effect of Adding the Mean Value
A data set has a mean of 20 and an SD of 5. If you add one new value equal to 20 to the set, what happens to the mean and the SD?
The mean remains exactly 20 because adding a value equal to the current mean does not change the total sum relative to the count. The SD decreases because the new point reduces the typical distance from the mean.
Watch out: Thinking the mean must change whenever a new number is added, or assuming the SD stays the same because the new value matches the center.
Sample questions
Question 1
Five numbers have a mean of 12. Four of them are 10, 14, 8, and 16. What is the fifth number?
- 10
- 48
- 12
- 14
Show answer and explanation
Answer: 12
The total must be 5 x 12 = 60; the four known values sum to 48, so the fifth is 12.
Question 2
In one class of 11 students, the mean test score was 62. In another class of 22 students, the mean score was 71. What was the mean score of all 33 students combined?
- 67
- 69
- 68
- 66.5
Show answer and explanation
Answer: 68
Total points = 11×62 + 22×71 = 2244. Mean = 2244/33 = 68.