Question

1. a policeman records the speeds of cars on a certain section of roadway with a radar gun. the histogram below shows the distribution of speeds for 251 cars. which of the following measures of center and spread would be the best ones to use when summarizing these data? (a) mean and interquartile range. (b) mean and standard deviation. (c) median and range. (d) median and standard deviation. (e) median and interquartile range.

Explanation:

Step1: Analyze the data distribution

The histogram shows a distribution of car speeds. We need to determine if the distribution is symmetric or skewed. From the histogram, the data seems to be roughly symmetric (bell - shaped), but we also consider the robustness of measures. However, when dealing with data that might have outliers (even if the main body is symmetric, extreme values at the tails can affect mean and standard deviation), but in a symmetric distribution, mean is a good measure of center and standard deviation (or interquartile range) for spread. Wait, no - actually, for a symmetric distribution without extreme outliers, mean and standard deviation are appropriate. But wait, let's recall: the mean is affected by outliers, while the median is robust. The standard deviation is also affected by outliers, and the interquartile range (IQR) is robust. But in a symmetric, unimodal distribution (like a normal - like distribution, which this histogram resembles), the mean and standard deviation are good measures. But wait, the options: let's check each option.

Option A: Mean and IQR. IQR is robust, but for a symmetric distribution, standard deviation is more appropriate as a measure of spread if there are no extreme outliers. But wait, maybe the distribution has some skewness? Wait, the histogram has a peak around 28 - 30, and tails on both sides. But the key is: when the distribution is symmetric, mean is a good center, and standard deviation (or IQR) for spread. But let's think about the measures:

- Mean: sensitive to outliers, but in a symmetric distribution without extreme outliers, it's good.
- Median: robust to outliers, used for skewed distributions.
- Standard deviation: sensitive to outliers, measures spread around mean.
- IQR: robust to outliers, measures spread of the middle 50% of data.
- Range: sensitive to outliers (max - min).

Now, the histogram of car speeds: it's a unimodal, roughly symmetric distribution (bell - shaped). So for a symmetric distribution, mean is a good measure of center, and standard deviation (or IQR) for spread. But let's check the options:

Option B: Mean and standard deviation. For a symmetric distribution, mean is the center, and standard deviation is the spread (since it measures how far data is from the mean, which is the center in symmetric case).

Wait, but maybe I made a mistake. Wait, the question is about the best measures. Let's re - evaluate:

If the distribution is symmetric, mean and standard deviation are appropriate. If it's skewed, median and IQR are better. The histogram here: let's see the left and right tails. The left tail (lower speeds) has low frequency, right tail (higher speeds) also low frequency. So it's roughly symmetric. So mean (center) and standard deviation (spread) would be appropriate. But wait, let's check the options again.

Wait, the options:

A: Mean and IQR - IQR is for robust spread, but mean is for symmetric.

B: Mean and standard deviation - correct for symmetric, no extreme outliers.

C: Median and range - range is bad (outlier - sensitive), median is for skewed.

D: Median and standard deviation - inconsistent (median is robust, standard deviation is not).

E: Median and IQR - for skewed distributions.

Since the histogram is roughly symmetric (bell - shaped), mean is a good center, and standard deviation is a good spread. So the answer should be B? Wait, no, wait. Wait, maybe the distribution is not perfectly symmetric, but let's think about the number of data points (251, which is large). Wait, maybe I misread the histogram. Wait, the x - axis is speed from 20 to 36, and the frequency increases to a peak and then decreases. So it's a unimodal, symmetric (or approximately symmetric) distribution. So in such cases, mean (center) and standard deviation (spread) are appropriate. So the correct option is B? Wait, no, wait. Wait, actually, in a symmetric distribution, mean and standard deviation are the preferred measures of center and spread. So the answer is B? Wait, but let's check again.

Wait, the key is: mean is affected by outliers, but if the distribution is symmetric, outliers are rare (since it's symmetric). So mean and standard deviation are appropriate. So the correct option is B: Mean and standard deviation.

Wait, but maybe I made a mistake. Let's recall: for a symmetric, unimodal distribution, mean and standard deviation are the standard measures. For skewed distributions, median and IQR are better. So since the histogram looks like a normal distribution (symmetric, unimodal), mean and standard deviation are appropriate. So the answer is B.

Answer:

B. Mean and standard deviation.