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One-variable data: Distributions and measures of center and spread: SAT Math notes
These questions test whether you can find and compare the mean, median, mode, range and standard deviation of a data set, read from a list, table or graph, and predict how they change when the data change.
Study it with worked examples and 8 practice questionsPractise One-variable data: Distributions and measures of center and spread questions
What this topic tests
A one-variable data set is a list of values for a single thing, like the ages of 20 students or the number of piglets in each litter. Measures of center (mean and median) tell you where the data sit. Measures of spread (range and standard deviation) tell you how scattered they are.
The data come in many forms: a plain list, a frequency table, a bar graph, a dot plot, a histogram or a box plot. Your first job is to read the data correctly, especially in a frequency table, where each value counts as many times as its frequency.
Harder questions often need very little arithmetic. They ask which measure changes when an outlier is removed, which set has the bigger mean or standard deviation, or what a missing value must be. Thinking about what each measure means is faster than calculating everything.
Key ideas
- Mean = sum of the values ÷ number of values. Turned around: sum = mean × number of values.
- Median = the middle value once the data are in order. With n values it's value number (n+1)/2. With an even count, average the two middle values.
- Mode = the most common value. Range = largest value minus smallest value.
- Standard deviation measures how far the values typically sit from the mean. Bunched-up data have a small standard deviation; spread-out data have a large one.
- An outlier pulls the mean toward it but barely moves the median. A long tail of high values makes the mean bigger than the median.
- In a symmetric data set, the mean and the median are equal.
- Adding the same number to every value shifts the mean and median by that number but leaves the range and standard deviation unchanged.
- A box plot shows the median, the quartiles, the minimum and the maximum. It never shows the mean.
The 4 kinds of One-variable data: Distributions and measures of center and spread question
Sorted from every One-variable data: Distributions and measures of center and spread question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Calculate a measure from the data
How to spot it. You're given data in a list, table or graph and asked directly for a mean, median, mode or range, or for a single value read off the display. Frequency tables and bar graphs are common, and some questions compare one measure across two groups.
How to answer it:
- Read the data carefully. In a frequency table, each value appears as many times as its frequency.
- For a median, put the values in order and count to the middle: value number (n+1)/2.
- For a mean, add all the values (value × frequency, for a table) and divide by how many values there are.
- For a range, subtract the smallest value from the largest.
The trap. Forgetting to sort before finding the median, or treating each row of a frequency table as a single value. In a frequency table you find the middle by counting through the frequencies, not by picking the middle row.
2. Work backward with the mean
How to spot it. A mean is given and something else is missing: a value in the list, a score that was removed, or the mean of two groups combined. Phrases like the mean of the remaining, or a data set made from A and B together, are the signal.
How to answer it:
- Turn every mean into a total: sum = mean × count.
- Write the totals before and after the change, or for each group.
- Subtract or add the totals to find the missing value, or divide the combined total by the combined count.
- Check that the answer makes sense. For example, a removed value that lowered the mean must be above the old mean.
The trap. Working with the means instead of the totals. When groups have different sizes, their combined mean isn't halfway between the two means. And the value that was removed isn't the difference of the means (6.7 minus 6.25); it's the difference of the totals.
3. Compare centers without computing everything
How to spot it. The question asks which is bigger, the mean or the median, how centers compare across two data sets, or what happens to one of them when a value is added or removed. The choices are statements like the mean of B is greater than the mean of A, or orderings like mode < median < mean.
How to answer it:
- Look at the shape. Is the data symmetric? Is there an outlier or a long tail on one side?
- Find the median by its position. That usually needs no arithmetic.
- Decide how the mean compares: an outlier or tail pulls the mean toward it, while the median stays near the middle.
- Only calculate the mean exactly if the choices are too close to judge.
The trap. Assuming an outlier moves the median a lot. Removing or adding one extreme value can leave the median exactly where it was while shifting the mean quite a bit. Also, a box plot never shows the mean, so a choice about means can't be confirmed from one.
4. Spread: standard deviation and range
How to spot it. The question mentions standard deviation or range, or asks which measure stays the same after the data are changed: every value shifted, one value corrected, or values pushed away from the middle.
How to answer it:
- Remember what spread means: how far the values sit from the center. Range looks only at the two ends; standard deviation looks at every value.
- To compare standard deviations, find the set whose values cluster most tightly around the middle (smallest) or sit farthest out (largest). You don't need the formula.
- For a change to the data, test each measure. Does the sum change (mean)? Does the middle value change (median)? Do the ends move (range)? Do the values move closer to or farther from the center (standard deviation)?
The trap. Thinking that shifting every value changes the spread. Adding 46 to every value moves the whole set, but the gaps between values stay the same, so the range and standard deviation don't change. Also, two sets with the same mean can have very different spreads.
Practise One-variable data: Distributions and measures of center and spread
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