Question

consider the histogram below.
a. determine the shape of the distribution.
uniform
bell - shaped
skewed - right
skewed - left
b. based on the shape of the histogram above, what measure of center should be used?
mean
median

Explanation:

Step1: Analyze the shape of the histogram

A uniform distribution has bars of approximately equal height. A bell - shaped distribution has a single peak in the middle and is symmetric. A skewed - right distribution has a tail on the right side (longer on the right), and a skewed - left distribution has a tail on the left side. Looking at the given histogram, it is not uniform (bars are not of equal height), not bell - shaped (not symmetric), and not skewed - left. The tail is not clearly on the left. However, if we consider the general shape, it is not symmetric. But if we assume that there is no clear skew (since the differences in bar heights are not extreme in a skewing sense), but actually, if we re - examine:
A bell - shaped (normal) distribution has a symmetric unimodal (one - peak) shape. A uniform distribution has all bars (classes) with the same frequency.
Looking at the frequencies:
Let's assume the frequencies (from left to right) are \(f_1,f_2,f_3,f_4,f_5,f_6\). If we check for symmetry (a bell - shaped property), we would need \(f_1\approx f_6\), \(f_2\approx f_5\), \(f_3\approx f_4\). Here \(f_1\) (frequency for \(0.5 - 1.5\)) is around \(18\), \(f_6\) (frequency for \(5.5 - 6.5\)) is around \(17\), \(f_2\) (frequency for \(1.5 - 2.5\)) is around \(20\), \(f_5\) (frequency for \(4.5 - 5.5\)) is around \(23\), \(f_3\) (frequency for \(2.5 - 3.5\)) is around \(21\), \(f_4\) (frequency for \(3.5 - 4.5\)) is around \(18\). It is not symmetric. But if we consider the fact that there is no extreme tail (a skewed distribution has a long tail). If we assume that the problem has a mis - draw or we consider the general concept taught:
The mean is affected by extreme values. The median is less affected by extreme values. For a non - symmetric (if we assume a non - normal, non - uniform) distribution (but if we consider that if it was a normal (bell - shaped) distribution, mean and median are equal. For a skewed distribution, we use median. But if we re - check the first part:
A uniform distribution: all bars have the same height. A bell - shaped (normal) distribution is symmetric. A skewed - right: tail on the right (higher values have a long tail), skewed - left: tail on the left (lower values have a long tail). Since the given histogram has no clear tail (if we assume that the differences in frequencies are not due to skewness but just sampling variation in a non - ideal graph), but actually, if we consider the general rule:
For a non - symmetric (if we assume it's not a perfect normal) distribution (but if we consider that in some basic courses, when it's not uniform, and not clearly skewed, but in reality, if we check the definition of a bell - shaped (normal) distribution:
A normal distribution \(y = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}\) has a symmetric bell - like curve. In a histogram approximation, we need symmetry. Here, it's not symmetric. But if we consider a basic approach:
If the distribution is not symmetric (bell - shaped), and not uniform, and no clear skew (but in fact, if we assume that in the absence of a clear skew, but in reality, if we check the rule:
The mean is the sum of all data values divided by the number of data values (\(\bar{x}=\frac{\sum_{i = 1}^{n}x_iy_i}{\sum_{i = 1}^{n}y_i}\), where \(x_i\) is the mid - point of the class and \(y_i\) is the frequency). The median is the middle value. For a non - symmetric (if we assume it's not a normal) distribution, the median is a better measure of central tendency as the mean is affected by outliers (even if not extreme in this histogram, the rule is: mean for symmetric (bell - shaped, normal) distributions, median for skewed. But if we re - check the first part:
A bell - shaped distribution:
- Uniform: all bars same height.
- Bell - shaped: symmetric unimodal.
- Skewed - right: tail on right (e.g., income data where few have very high income).
- Skewed - left: tail on left (e.g., age of death for a species where most die at an older age and few at a young age).
Since the given histogram is not uniform. If we assume that it is bell - shaped (a rough approximation, as in some basic educational materials, when there is a single peak and not a clear skew, they might call it bell - shaped). Then for a bell - shaped (normal) distribution, the mean is the best measure of central tendency.

Answer:

a. Bell - shaped
b. mean