Question

1. given a set of data: 265,263,269,259,267,264,253,275,264,260,273,257,and 291.

\ta. make a stem - and - leaf plot of this data.
\tb. find the mean and median of this data.
\tc. find the range of this data.
\td. make a box plot for this data.

1. given a set of data: 48,42,37,29,49,46,38,28,45,45,35,46.25,34,46,46.5,43,46.5,48,41.25,29,and 47.75.

\ta. make a stem - and - leaf plot of this data.
\tb. find the mean and median of this data.
\tc. find the range of the data.
\td. make a box plot for this data.

Explanation:

15.

Step1: Arrange data for stem - and - leaf plot

First, arrange the data in ascending order: 253, 257, 260, 263, 264, 264, 265, 267, 269, 273, 275, 291. The stem - and - leaf plot:
Stem | Leaf
--- | ---
25 | 3 7
26 | 0 3 4 4 5 7 9
27 | 3 5
29 | 1

Step2: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 12$ and $\sum_{i=1}^{12}x_{i}=253 + 257+260+263+264+264+265+267+269+273+275+291=3241$. So, $\bar{x}=\frac{3241}{12}\approx270.08$.

Step3: Calculate the median

Since $n = 12$ (even), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered values. The 6th and 7th values are 264 and 265. Median $=\frac{264 + 265}{2}=264.5$.

Step4: Calculate the range

The range is the difference between the maximum and minimum values. Range $=291-253 = 38$.

Step5: Make a box - plot

First, find the first quartile $Q_1$. The lower half of the data is 253, 257, 260, 263, 264, 264. The median of this lower half (for $n = 6$) is $\frac{260+263}{2}=261.5$. The third quartile $Q_3$. The upper half of the data is 265, 267, 269, 273, 275, 291. The median of this upper half (for $n = 6$) is $\frac{269+273}{2}=271$. The box - plot has a minimum of 253, $Q_1 = 261.5$, median = 264.5, $Q_3=271$, and maximum of 291.

Step1: Arrange data for stem - and - leaf plot

Arrange the data in ascending order: 28, 29, 29, 34, 35, 37, 38, 41.25, 42, 43, 45, 45, 46, 46.25, 46.5, 46.5, 47.5, 48, 48, 49. The stem - and - leaf plot:
Stem | Leaf
--- | ---
2 | 8 9 9
3 | 4 5 7 8
41 | 25
42 |
43 |
45 | 5
46 | 25 5 5
47 | 5
48 | 8
49 |

Step2: Calculate the mean

$n = 20$, $\sum_{i = 1}^{20}x_{i}=28+29+29+34+35+37+38+41.25+42+43+45+45+46+46.25+46.5+46.5+47.5+48+48+49 = 774$. Mean $\bar{x}=\frac{774}{20}=38.7$.

Step3: Calculate the median

Since $n = 20$ (even), the median is the average of the 10th and 11th ordered values. The 10th value is 43 and the 11th value is 45. Median $=\frac{43 + 45}{2}=44$.

Step4: Calculate the range

Range $=49 - 28=21$.

Step5: Make a box - plot

The lower half of the data is 28, 29, 29, 34, 35, 37, 38, 41.25, 42, 43. The median of this lower half (for $n = 10$) is $\frac{35+37}{2}=36$. The upper half of the data is 45, 45, 46, 46.25, 46.5, 46.5, 47.5, 48, 48, 49. The median of this upper half (for $n = 10$) is $\frac{46.5+46.5}{2}=46.5$. The box - plot has a minimum of 28, $Q_1 = 36$, median = 44, $Q_3=46.5$, and maximum of 49.

Answer:

a. Stem - and - leaf plot: Stem | Leaf; 25 | 3 7; 26 | 0 3 4 4 5 7 9; 27 | 3 5; 29 | 1
b. Mean $\approx270.08$, Median = 264.5
c. Range = 38
d. Box - plot: Min = 253, $Q_1 = 261.5$, Median = 264.5, $Q_3=271$, Max = 291

16.