EQN-06-07

Comparison with Averages & Data

Compare means, medians, and totals using the sum identity instead of recomputing everything.

~5 min read•Difficulty 2/5

Use the sum identity

The key tool stays: sum = mean × count. Many average comparisons are settled by comparing totals, not by re-finding each mean.

A : mean of 10 numbers whose sum is 200 \to 20

B : 18

A is greater .

Adding a value moves the mean predictably

Add a value above the current mean $\to$ the mean rises.
Add a value equal to the mean $\to$ the mean is unchanged.
Add a value below the mean $\to$ the mean falls.

This alone answers many "new average" comparisons without arithmetic.

Mean vs median

For a symmetric set they are close; a single large outlier pulls the mean up but barely moves the median. If a set has an extreme high value, expect mean $>$ median.

When data are incomplete

If the set is only partly described (an unknown value, an open range), different fillings can change the order $\to$ choice D.

Worked examples

Example 1 (difficulty 1/5)

A set of 6 numbers has a sum of 90. Quantity A: the mean. Quantity B: 14. Which is greater?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A list of 5 numbers has mean 10. A sixth number, 10, is added. Quantity A: the new mean. Quantity B: 10. Which is greater?

◎ Predict the answer first:

Tips

Reach for sum $=$ mean $\times$ count before doing heavy arithmetic.
Compare a new value to the current mean to predict whether the mean rises, holds, or falls.
A big outlier raises the mean far more than the median.

Common mistakes

Recomputing every value instead of using totals

Why: It feels safer to compute fully.

Fix: Comparisons of means usually need only the sums and counts.

Assuming mean equals median

Why: They coincide for symmetric data.

Fix: Outliers separate them; the mean follows the tail.

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