Transformations of data sets flashcards

Topic 6: Transformations of data sets

Each value x is coded as y = ax + b. What happens to the mean and the median?

Both follow the whole rule: new mean = a × mean + b, new median = a × median + b.

Each value x is coded as y = ax + b. What happens to the standard deviation and the IQR?

Both are multiplied by |a|. Spread measures ignore b and ignore the sign of a, so a spread is never negative.

Data with mean 30 and standard deviation 4 are each multiplied by 2.5. Find the new mean and standard deviation

Mean = 30 × 2.5 = 75. Standard deviation = 4 × 2.5 = 10. (Adding 12 to every value instead would give mean 42 and standard deviation 4.)

A data set with median 55 and IQR 14 is coded with y = 100 − x. Find the new median and IQR

Median = 100 − 55 = 45. IQR = |−1| × 14 = 14. The order reverses, so the old upper quartile becomes the new lower quartile.

A value equal to the mean is added to a data set. What happens to the mean and the standard deviation?

The mean stays the same. The standard deviation decreases, because n rises while the sum of squared deviations does not change.

A value above the median is added to a data set. What can you say about the median?

It increases or stays the same; it cannot decrease.

How do you find the combined standard deviation of two groups?

For each group, total = n × mean and Σx² = n × (σ² + mean²). Add the totals, the Σx² values and the group sizes. Combined variance = (Σx² of both) ÷ (n₁ + n₂) − (combined mean)²; then square root.

A group of 20 values has mean 5 and a group of 30 values has mean 10. Find the combined mean

(20 × 5 + 30 × 10) ÷ 50 = 400 ÷ 50 = 8.

What is a z-score (standardised value)?

z = (x − mean) ÷ σ: the number of standard deviations a value lies above or below the mean. Standardised data have mean 0 and standard deviation 1, so z-scores compare results from different distributions.

A mark of 70 comes from a test with mean 58 and σ = 8. Scale it to mean 50 and σ = 10

z = (70 − 58) ÷ 8 = 1.5. New value = new mean + new σ × z = 50 + 10 × 1.5 = 65.

How do you compare two distributions in a full answer?

Give one comparison of centre (mean or median: which has larger values generally) and one of spread (standard deviation or IQR: smaller means more consistent), with the figures and the context. Pair mean with SD, median with IQR.