Question

The infant mortality rate is defined as the number of infant deaths per 1,000 live births. This rate is often used as an indicator of the level of health in a country. The relative frequency histogram below shows the distribution of estimated infant death rates for 224 countries for which such data were available in 2014.31
(a) Estimate Q1, the median, and Q3 from the histogram.
(b) Would you expect the mean of this data set to be smaller or larger than the median? Explain your reasoning.
The graph that goes with it is in the picture.

Answer 1

(a)

Q1, the first quartile, 25th percentile, is greater than or equal to 1/4 of the points.  It's in the first bar so we can estimate Q1=5.   In reality the bar includes values from 0 to 9 or 10  (not clear which) and has around 37% of the points so we might estimate Q1 a bit higher as it's 2/3 of the points, say Q1=7.

The median is bigger than half the points.  First bar is 37%, next is 22%, so its about halfway in the second bar, median=15

Third bar is 11%, so 70% so far.  Four bar is 5%, so we're at the right end of the fourth bar for Q3, the third quartile, 75th percentile, say Q3=40

b

When the data is heavily skewed left like it is here, the median tends to be lower than the mean.  The 5% of the data from 80 to 120 averages around 100 so adds 5 to the mean, and 8% of the data from the 60 to 80 adds another 5.6, 15% of the data from 40 to 60 adds about 7.5, plus the rest, so the mean is gonna be way bigger than the median of around 15.