One-variable data: distributions and measures of center and spread
Problem-Solving and Data Analysis is about 15% of SAT Math (5–7 of 44 questions). One-variable data questions ask you to calculate or compare the mean and median, say how an outlier or a new value changes them, compare spread, and read a table or chart.
Center: mean, median and mode
The mean is the sum of the values divided by their number. The median is the middle value after you put the values in order; with an even number of values it is the average of the two middle ones. The mode is the value that occurs most often. For the data 4, 5, 5, 7, 9 the sum is 30, so the mean is 30 ÷ 5 = 6. The median is 5 and the mode is 5. The mean is the balance point of the data, so it is pulled toward very large or very small values.
Spread: range and standard deviation
Two data sets can have the same mean and look very different. The range is the largest value minus the smallest, so it uses only two values. The standard deviation uses every value: it tells you, roughly, how far the values typically are from the mean. In the table, set B has the larger range and the larger standard deviation, because its values are far from the mean of 50; set A is bunched close to the mean, so its standard deviation is small. On the SAT you compare standard deviations; you do not calculate them by hand.
| Set | Values | Mean | Range |
|---|---|---|---|
| A | 48, 49, 50, 51, 52 | 50 | 4 |
| B | 30, 40, 50, 60, 70 | 50 | 40 |
What an outlier does
An outlier changes some measures a lot and others very little. The mean, the range and the standard deviation all react strongly. The median hardly moves, because it depends on the middle of the ordered list, not on the extreme values. For this reason the median describes a typical value better when the data have an outlier, as with house prices or salaries.
| Measure | 12, 13, 14, 15, 16 | Same data plus 40 |
|---|---|---|
| Mean | 14 | about 18.3 |
| Median | 14 | 14.5 |
| Range | 4 | 28 |
| Standard deviation | small | much larger |
Reading histograms and box plots
A histogram groups the data into intervals. The height of each bar is the number of values in that interval, so you cannot read exact values from it, but you can add the bars to find the total and use running totals to find which interval holds the median. In the histogram below, 4 + 9 = 13 of the 20 values are at most 5, so the median (between the 10th and 11th values) is in the interval 3–5. A box plot shows five numbers: the minimum, Q1, the median, Q3 and the maximum. Each of the four parts holds about 25% of the data. The box shows the middle half, and the interquartile range is IQR = Q3 − Q1.
Frequency tables and combined means
A frequency table lists each value and how many times it occurs. To find the mean, multiply each value by its frequency, add these products, and divide by the total frequency. In the table below the mean is (0·4 + 1·6 + 2·7 + 3·3) ÷ 20 = 29 ÷ 20 = 1.45. The median of these 20 values is the average of the 10th and 11th values, which are 1 and 2, so it is 1.5. To combine the means of two groups, weight each mean by the size of its group. Never just average the two means unless the groups have equal sizes.
| Books read | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Students | 4 | 6 | 7 | 3 |
A quick formula for combined means
If group 1 has n₁ values with mean m₁ and group 2 has n₂ values with mean m₂, the total of all values is n₁m₁ + n₂m₂. Dividing by the total number of values gives the combined mean. The same idea works backward: if you know the mean and the number of values, you know the total, and you can see what a new value does to it.
- Mean: (14 + 9 + 21 + 9 + 12) ÷ 5 = 65 ÷ 5 = 13.
- Median: put the numbers in order: 9, 9, 12, 14, 21. The middle one is 12.
- Mode: 9 occurs twice, more often than any other number.
- The mean increases by more than the median does.
- The median increases by more than the mean does.
- The mean and the median both stay the same.
- The range stays the same.
- Before: the mean is 1,000 ÷ 5 = 200 and the median is 200.
- After: the mean is 1,900 ÷ 6, about 316.7; the median is the average of 200 and 210, which is 205.
- The mean rose by about 116.7 and the median by only 5. The range jumped from 40 to 720.
| Class | Students | Mean score |
|---|---|---|
| A | 20 | 78 |
| B | 30 | 88 |
- Total for class A: 20 × 78 = 1,560.
- Total for class B: 30 × 88 = 2,640.
- Combined: (1,560 + 2,640) ÷ 50 = 4,200 ÷ 50 = 84.
- With 40 values, the median is the average of the 20th and 21st values.
- Running totals: 4 in the first interval, 4 + 9 = 13 in the first two, 13 + 12 = 25 in the first three.
- The 20th and 21st values are both beyond 13 and within 25, so both are in the third interval.
- Not putting the data in orderThe median is the middle of the sorted list. Sort first, then count in from both ends.
- Forgetting to average the two middle valuesWith an even number of values there is no single middle. 3, 4, 8, 9 has median (4 + 8) ÷ 2 = 6.
- Averaging the means of unequal groupsMultiply each mean by its group size, add the totals, and divide by the total number of values.
- Thinking the mean ignores outliersThe mean follows the outlier. The median is the measure that stays near the middle.
- Mixing up the mode and its frequencyThe mode is the value that occurs most often, not the number of times it occurs.
- Reading a histogram bar as a data valueThe height of a bar is a count of values in the interval. Add the bars to get the total number of values.
- Confusing size with spreadA data set with large numbers does not have a large standard deviation. Spread is about distance from the mean, not about how big the numbers are.
Desmos can do the arithmetic for a list. Make a list by typing L = [14, 9, 21, 9, 12]. Then type mean(L) and median(L) on new lines; Desmos shows the values. You can also type stdev(L), min(L) and max(L). The commands histogram(L), dotplot(L) and boxplot(L) draw the displays. For a frequency table, make two lists, V for values and F for frequencies, and type total(V * F) / total(F) for the mean.
- Type L = [14, 9, 21, 9, 12]
- Type mean(L) to see 13
- Type median(L) to see 12
- Type boxplot(L) to see the five-number summary as a picture
Use Desmos for long lists or to check a mean. For five or six numbers, do it by hand. Remember that the SAT asks you to compare standard deviations rather than compute them.
Open Desmos ↗