Question

which measure would be the most appropriate to describe the center of the data in the histogram below? daily high temperatures (with a histogram showing number of days on the y - axis and high temperatures in ranges 50 - 54, 55 - 59, 60 - 64, 65 - 69, 70 - 74 on the x - axis) options: mean, range, median, interquartile range

Explanation:

Step1: Analyze the Histogram Shape

The histogram shows a skewed or non - symmetric distribution? Wait, actually, the main cluster is at 65 - 69, and the other bars are smaller. But when data is not strongly skewed or has a single peak, but let's recall the measures of center. Mean is affected by outliers or skewness. Median is a resistant measure. But first, let's recall what each measure is:
- Mean: Average, sum of values divided by number of values.
- Range: Measure of spread (max - min), not center.
- Median: Middle value when data is ordered.
- Interquartile range (IQR): Measure of spread (Q3 - Q1), not center.

So we can eliminate range and IQR as they are measures of spread, not center. Now between mean and median. For a distribution that is not extremely skewed, but let's check the data. The histogram has a peak at 65 - 69, and the other groups have low frequencies. The median is a better measure of center when the data has some skewness or when there are extreme values (but here the non - peak groups are not extreme, but the key is that range and IQR are spread, so we eliminate them. Then between mean and median, but since range and IQR are out, and we need center, so median or mean. Wait, no: range is max - min (spread), IQR is Q3 - Q1 (spread). So the options for center are mean and median. But wait, the question is about center. Wait, the options: mean (center), range (spread), median (center), interquartile range (spread). So first, eliminate the spread measures (range and IQR). Now, for a distribution that is symmetric, mean and median are close. But if the distribution is skewed, median is better. But in this histogram, the data is somewhat skewed left? Wait, the left side (50 - 54, 55 - 59, 60 - 64) has low frequencies, and the main peak is at 65 - 69, then a small peak at 70 - 74. But actually, the median is a measure of center, and since we have to choose between mean and median (and the other two are spread), but wait, no: the question is which is most appropriate to describe the center. Wait, maybe I made a mistake. Wait, range and IQR are spread, so they can't be center. So we have mean and median. But let's think again. The median is a better measure of center when the data has outliers or is skewed. But in this case, the data is not extremely skewed, but the key is that range and IQR are not measures of center. So we can eliminate B (range) and D (interquartile range). Now between A (mean) and C (median). Wait, but maybe the data is such that the median is more appropriate? Wait, no, maybe I messed up. Wait, the histogram: let's count the number of days. Let's assume the heights: 50 - 54: 1 (approx), 55 - 59: 3, 60 - 64: 3, 65 - 69: 19, 70 - 74: 3. Total days: 1+3 + 3+19+3=29. The median is the 15th value (since 29 is odd, (29 + 1)/2 = 15th). Let's order the data: the first group (50 - 54) has 1, 55 - 59 has 3 (total 4), 60 - 64 has 3 (total 7), 65 - 69 has 19 (total 7+19 = 26), 70 - 74 has 3 (total 29). So the 15th value is in the 65 - 69 group. The mean would be calculated as (sum of (midpoint * frequency))/total. Midpoints: 52, 57.5, 62, 67, 72. Frequencies: 1, 3, 3, 19, 3. Sum = 52*1+57.5*3 + 62*3+67*19+72*3=52 + 172.5+186+1273+216=52+172.5 = 224.5; 224.5+186 = 410.5; 410.5+1273 = 1683.5; 1683.5+216 = 1899.5. Mean = 1899.5/29≈65.5. The median is in the 65 - 69 group, so around 67 (midpoint). But the key is that the question is about which measure is most appropriate. Wait, but maybe the distribution is not symmetric, but the median is a better measure? Wait, no, maybe I made a mistake in eliminating. Wait, the options: range and IQR are spread, so they are out. So between mean and median. But actually, the median is a measure of center, and for a distribution with a single peak (unimodal) and not extremely skewed, both can be used, but maybe the question is trying to trick us into realizing that range and IQR are spread. Wait, no, the question is "measure to describe the center", so range (spread) and IQR (spread) are out. So we have mean and median. But maybe the answer is median? Wait, no, maybe I made a mistake. Wait, let's re - read the options:
- mean: center
- range: spread
- median: center
- interquartile range: spread

So we need to choose between mean and median. But in a distribution that is skewed, median is better. But in this case, the data is somewhat skewed left (the left tail has low frequencies). So median is a better measure of center than mean when there is skewness. But wait, maybe the answer is median? Wait, no, maybe I'm wrong. Wait, let's think again. The histogram has a main peak at 65 - 69, and the other bars are small. The median is the middle value, which is less affected by the small number of days in the lower temperature ranges. So the most appropriate measure of center here is the median. Wait, but why? Because the mean would be pulled a bit by the lower temperatures, but the median is in the main peak. Alternatively, maybe the answer is median. Wait, but let's check the options again. The options are mean, range, median, interquartile range. So we can eliminate range and interquartile range (since they are spread). Then between mean and median. For a distribution with a single peak (unimodal) and no extreme skewness, but in this case, the left side is a bit skewed. So median is more appropriate. So the answer is median.

Answer:

C. median