Comparing distributions · SAT Math

The mean uses every observation, the median uses ordered position, and spread measures variation. A frequency table weights a value by how often it occurs. A boxplot marks a five-number summary; tied observations and quartile conventions mean one should not automatically assign exactly one quarter of a small data set to every drawn segment.

Combine totals rather than averages of averages

Three measurements with mean ten have total 30. Seven measurements with mean twenty have total 140. Together their mean is 170/10=17, not 15. The unweighted average of two means is guaranteed to work for equal group sizes; unequal sizes require the observation counts as weights.

Transform location and spread

If every value z becomes 3z−7, each deviation from the new mean is three times its old deviation. Standard deviation triples; subtracting seven does not affect it. Variance instead scales by nine. Confusing those two summaries is a different error from applying the translation to a spread.

Use positions and quartiles

If values two, five and ten occur with frequencies two, five and three, the fifth and sixth observations are both five, so the median is five. For a boxplot with Q1=4 and Q3=15, the IQR is eleven. The minimum and maximum would be used for the full range, not the IQR.

Transfer and check

A data set with standard deviation two is transformed by z→4z+9. Its new standard deviation is eight. Explain why the added nine matters for the mean but not the deviation from the mean. Then give a pair of data sets with equal means and different spreads, such as 4,4,4 and 0,4,8.

WORKED EXAMPLE

One group contains 3 measurements with mean 11. A second contains 7 measurements with mean 21. What is the mean of all the measurements together?

The group totals are 33 and 147.
There are 10 observations altogether.
Divide the total by the combined count: 18.

Answer: 18

Compare the other answer choices

Attach each group size to the other group mean.

Average the two means without weighting unequal group sizes.

180

Return the combined total rather than dividing by total count.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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