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One-Variable Data: Distributions and Measures of Center and Spread
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Order first, then count to the middle
- Start:
- The list of numbers, before anything else.
- Stop:
- When the numbers are in order and you have counted the middle position.
- Skip:
- Any calculation of the mean if only the median is asked.
- Why:
- The median is defined on ordered data, so ordering first prevents the most common mistake.
Used in questions 2, 4
2. Sum = mean × count
- Start:
- The sentence that gives a mean and how many values there are.
- Stop:
- When you have the total sum written down.
- Skip:
- Trying to guess the missing number from the mean directly.
- Why:
- Missing-value and added-value questions are solved by comparing two sums, which is always safe.
Used in questions 3, 5, 8
3. Frequency table: running total
- Start:
- The frequency column, added from the top.
- Stop:
- When the running total passes the middle position (for 20 values, positions 10 and 11).
- Skip:
- The row labels in the middle of the table; they are not the median.
- Why:
- The median is a position in the data, so you find it by counting positions.
Used in question 4
4. Compare standard deviations by distance from the mean
- Start:
- The mean of each set.
- Stop:
- When you can say which set sits farther from its mean.
- Skip:
- Calculating the standard deviation. The SAT never needs the number.
- Why:
- Standard deviation is the typical distance from the mean, so wider-looking data has the larger one.
Used in question 6
5. Outlier rule
- Start:
- The new value that is added or removed.
- Stop:
- When you know whether it is far from the others.
- Skip:
- Recalculating the median fully; it moves only a little.
- Why:
- An extreme value changes the sum and the range a lot but only shifts the middle position by one step.
Used in question 7
6. Weighted mean by distances
- Start:
- The two group means and the combined mean.
- Stop:
- When you have the distance of each group mean from the combined mean.
- Skip:
- Averaging the two means directly.
- Why:
- The group closer to the combined mean is bigger, in the opposite ratio of the distances. Then check with the sums.
Used in question 8
7. Calculator shortcut
- Start:
- Desmos: type mean( or median( with a list in square brackets.
- Stop:
- When the number appears; compare it with the options.
- Skip:
- Adding long lists by hand.
- Why:
- Typing mean([3,5,6,9,12]) and median([2,4,5,7,9,12,15]) is quick and avoids arithmetic slips.
Used in questions 1, 2, 5
8. Graph: count dots or bars from the left
- Start:
- The axis labels, then the dots or bars at the left end.
- Stop:
- When your running total reaches the middle position (for 15 values, the 8th; for 35, the 18th).
- Skip:
- The tallest bar or the tallest stack. It shows the mode, not the median.
- Why:
- The median is a position in the ordered data, and a dot plot or histogram is already in order from left to right.
Used in questions 10, 12
9. Frequency table: value × frequency, then add, then divide by the total frequency
- Start:
- The two columns: the value and how many times it occurs.
- Stop:
- When you have the sum of (value × frequency) and the sum of the frequencies.
- Skip:
- Dividing by the number of rows. The rows are not the data points.
- Why:
- Each row stands for many data points, so the mean must count each row as often as its frequency says.
Used in questions 9, 14, 16
10. Multiply changes spread, adding does not
- Start:
- The sentence that says how every value is changed.
- Stop:
- When you know which operation is multiplying and which is adding or subtracting.
- Skip:
- Recalculating the standard deviation from scratch.
- Why:
- Sliding all values by the same amount moves the center but keeps the distances between them. Multiplying stretches those distances.
Used in question 13