Question

four distributions, labeled (a), (b), (c), and (d) are represented below by their histograms. each distribution is made of 9 measurements. without performing any calculations, order their respective means $mu_a$, $mu_b$, $mu_c$, and $mu_d$.
enter the four subscripts appropriately below.
(histograms for (a), (b), (c), (d) are shown with axes from 1 to 14)
$mu_{square} < mu_{square} < mu_{square} < mu_{square}$

Explanation:

To order the means, we analyze the histograms by looking at the spread and central tendency. The mean is affected by the position of the data (higher - valued bins contribute more to the mean).

Step 1: Analyze Distribution (b)

Distribution (b) has data more concentrated on the left (lower values) and some on the right but the left - side concentration is strong. The lower - valued bins (2, 3) have a significant number of measurements.

Step 2: Analyze Distribution (d)

Distribution (d) has data in bins 2 - 3, 6 - 7, and 10 - 11. The bin 6 - 7 is a middle - range bin, so it pulls the mean a bit higher than (b) since it has data in a higher - valued bin than the left - most of (b) without a very low - valued offset.

Step 3: Analyze Distribution (a)

Distribution (a) has data in bins 2 - 3, 6, and 9 - 11. The bin 9 - 11 is a relatively high - valued range. The presence of data in 9 - 11 (higher than 6 - 7 in (d)) will pull the mean higher than (d).

Step 4: Analyze Distribution (c)

Distribution (c) has data in bins 2 - 3, 6, and 9 - 11, but the bin 10 - 11 has a more significant presence (or the data in the higher - valued bins is more weighted) compared to (a). So the mean of (c) is the highest among the four.

Answer:

$\mu_{b}<\mu_{d}<\mu_{a}<\mu_{c}$