Question

1. the following are the descriptive statistics of the previous histogram. use them to answer the following questions.

descriptive statistics:
sample size mean std. dev. min. q1 median q3 max. iqr
253 6.26 2.92 1 4 6 8 18 4
a. identify the mean and interpret it in a sentence. use the proper notation.
b. identify the median and interpret in a sentence.
c. when choosing the center that best describes the histogram, which do you choose? explain.
d. identify the standard deviation and interpret in a sentence. use the proper notation.
e. there is one value of 18, is this value unusual? explain.

Explanation:

Step1: Identify the mean

The mean is given as $\bar{x}=6.26$. It represents the average value of the data - set. On average, the values in the data - set are around 6.26.

Step2: Identify the median

The median is 6. This means that when the data values are arranged in ascending order, the middle - valued observation (or the average of the two middle - valued observations for an even - sized data set) is 6. Half of the data values are less than or equal to 6 and half are greater than or equal to 6.

Step3: Choose the best measure of center

The median is a better measure of the center. The presence of the maximum value 18 (which is relatively far from the other values) indicates that the data may be skewed. The mean is affected by extreme values, while the median is not. So, the median gives a more representative measure of the center for skewed data.

Step4: Identify the standard deviation

The standard deviation is $s = 2.92$. It measures the amount of variation or dispersion in the data - set. The values in the data - set typically vary from the mean by about 2.92 units.

Step5: Determine if 18 is unusual

We can use the range - rule of - thumb for significant values. Significant high values are $\mu+2\sigma$ and significant low values are $\mu - 2\sigma$. Here, $\mu = 6.26$ and $\sigma=2.92$. $\mu+2\sigma=6.26 + 2\times2.92=6.26+5.84 = 12.1$. Since $18>12.1$, the value 18 is an unusual value.

Answer:

a. The mean is $\bar{x}=6.26$. On average, the values in the data - set are around 6.26.
b. The median is 6. Half of the data values are less than or equal to 6 and half are greater than or equal to 6.
c. The median is a better measure of the center. The data may be skewed due to the extreme value of 18, and the median is not affected by extreme values as much as the mean.
d. The standard deviation is $s = 2.92$. The values in the data - set typically vary from the mean by about 2.92 units.
e. The value 18 is unusual. Using the range - rule of - thumb with $\mu = 6.26$ and $\sigma = 2.92$, significant high values are $\mu+2\sigma=12.1$ and $18>12.1$.