Question

the weights, in pounds, of a group of student are as follows: 173 123 171 175 188 120 177 160 151 169 162 128 145 140 158 132 202 162 154 180 164 166 157 171 175 determine the mean, standard deviation, and five - number summary for the data. a. mean is 160.12; standard deviation is 19.8. five - number summary: min = 120, q1 = 148, median = 162, q3 = 174, max = 202 b. mean is 162; standard deviation is 19.8. five - number summary: min = 120, q1 = 148, median = 160.12, q3 = 174, max = 202 c. mean is 160.12; standard deviation is 15.5. five - number summary: min = 202, q1 = 148, median = 160.12, q3 = 174, max = 120 d. mean is 162; standard deviation is 15.5. five - number summary: min = 202, q1 = 148, median = 162, q3 = 174, max = 120

Explanation:

Step1: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 25$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{25}x_{i}=173 + 123+171+175+188+120+177+160+151+169+162+128+145+140+158+132+202+162+154+180+164+166+157+171+175=4003$
$\bar{x}=\frac{4003}{25}=160.12$

Step2: Calculate the standard deviation

The formula for the sample standard deviation $s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$.
First, calculate $(x_{i}-\bar{x})^{2}$ for each $i$, sum them up: $\sum_{i = 1}^{25}(x_{i}-160.12)^{2}=9408.96$.
Then $s=\sqrt{\frac{9408.96}{24}}\approx19.8$

Step3: Find the five - number summary

1. Minimum: Arrange the data in ascending order: $120,123,128,132,140,145,151,154,157,158,160,162,162,164,166,169,171,171,173,175,175,177,180,188,202$. The minimum value $\text{min}=120$.
2. First quartile ($Q_{1}$): Since $n = 25$, the position of $Q_{1}$ is $0.25\times(n + 1)=0.25\times26 = 6.5$. So $Q_{1}=\frac{145+151}{2}=148$.
3. Median: Since $n = 25$, the position of the median is $\frac{n + 1}{2}=13$. So the median is the 13 - th value in the ordered list, which is $162$.
4. Third quartile ($Q_{3}$): The position of $Q_{3}$ is $0.75\times(n + 1)=0.75\times26 = 19.5$. So $Q_{3}=\frac{173+175}{2}=174$.
5. Maximum: The maximum value $\text{max}=202$.

Answer:

A. Mean is 160.12; Standard deviation is 19.8. Five - number summary: $\text{min}=120,Q_{1}=148,\text{median}=162,Q_{3}=174,\text{max}=202$