Question

1. answer each question using the list below 4, 5, 8, 1, 2, 3, 0, 8, 7, 6, 3

mean =
median =
standard deviation =
range =

1. answer each question using the list below 23, 10, 11, 24, 32, 15, 13, 16

mean =
median =
standard deviation =
range =

1. answer each question using the list below 3123, 1040, 1511, 2124, 1332, 2154, 5132, 6160

mean =
median =
standard deviation =
range =

1. answer each question using the list below 4, 8, 6, 10, 16, 14, 12, 18

mean =
median =
standard deviation =
range =

Explanation:

Step 1: Calculate mean for first list (4, 5, 8, 1, 2, 3, 0, 8, 7, 6, 2)

Sum all numbers: $4 + 5+8 + 1+2 + 3+0 + 8+7 + 6+2=46$. There are $n = 11$ numbers. Mean $\bar{x}=\frac{46}{11}\approx4.18$

Step 2: Calculate median for first list

Arrange in ascending - order: $0,1,2,2,3,4,5,6,7,8,8$. The median is the 6th number, so median $ = 4$

Step 3: Calculate standard deviation for first list

First, find the variance. Calculate the squared - differences from the mean for each number, sum them, and divide by $n$.
\[ \begin{align*} \sum_{i = 1}^{n}(x_i-\bar{x})^2&=(0 - 4.18)^2+(1 - 4.18)^2+(2 - 4.18)^2+(2 - 4.18)^2+(3 - 4.18)^2+(4 - 4.18)^2+(5 - 4.18)^2+(6 - 4.18)^2+(7 - 4.18)^2+(8 - 4.18)^2+(8 - 4.18)^2\\ &=17.47+9.90+4.75+4.75+1.39+0.03+0.67+3.31+8.06+14.59+14.59\\ &=79.41 \end{align*} \]
Variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}=\frac{79.41}{11}\approx7.22$. Standard deviation $s=\sqrt{7.22}\approx2.69$

Step 4: Calculate range for first list

Range = maximum value - minimum value. Maximum value $ = 8$, minimum value $ = 0$. Range $=8 - 0 = 8$

Step 5: Calculate mean for second list (23, 10, 11, 24, 32, 15, 13, 16)

Sum all numbers: $23+10 + 11+24+32+15+13+16=144$. There are $n = 8$ numbers. Mean $\bar{x}=\frac{144}{8}=18$

Step 6: Calculate median for second list

Arrange in ascending - order: $10,11,13,15,16,23,24,32$. The median is the average of the 4th and 5th numbers. Median $=\frac{15 + 16}{2}=15.5$

Step 7: Calculate standard deviation for second list

\[ \begin{align*} \sum_{i = 1}^{n}(x_i-\bar{x})^2&=(10 - 18)^2+(11 - 18)^2+(13 - 18)^2+(15 - 18)^2+(16 - 18)^2+(23 - 18)^2+(24 - 18)^2+(32 - 18)^2\\ &=64+49+25+9+4+25+36+196\\ &=408 \end{align*} \]
Variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}=\frac{408}{8}=51$. Standard deviation $s=\sqrt{51}\approx7.14$

Step 8: Calculate range for second list

Maximum value $ = 32$, minimum value $ = 10$. Range $=32 - 10 = 22$

Step 9: Calculate mean for third list (3123, 1040, 1511, 2124, 1332, 2154, 5132, 6160)

Sum all numbers: $3123+1040+1511+2124+1332+2154+5132+6160 = 22576$. There are $n = 8$ numbers. Mean $\bar{x}=\frac{22576}{8}=2822$

Step 10: Calculate median for third list

Arrange in ascending - order: $1040,1332,1511,2124,2154,3123,5132,6160$. Median $=\frac{2124 + 2154}{2}=2139$

Step 11: Calculate standard deviation for third list

\[ \begin{align*} \sum_{i = 1}^{n}(x_i-\bar{x})^2&=(1040 - 2822)^2+(1332 - 2822)^2+(1511 - 2822)^2+(2124 - 2822)^2+(2154 - 2822)^2+(3123 - 2822)^2+(5132 - 2822)^2+(6160 - 2822)^2\\ &=(- 1782)^2+(-1490)^2+(-1311)^2+(-698)^2+(-668)^2+(301)^2+(2310)^2+(3338)^2\\ &=3175524+2220100+1718721+487204+446224+90601+5336100+11142244\\ &=24516728 \end{align*} \]
Variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}=\frac{24516728}{8}=3064591$. Standard deviation $s=\sqrt{3064591}\approx1750.60$

Step 12: Calculate range for third list

Maximum value $ = 6160$, minimum value $ = 1040$. Range $=6160 - 1040 = 5120$

Step 13: Calculate mean for fourth list (4, 8, 6, 10, 16, 14, 12, 18)

Sum all numbers: $4+8+6+10+16+14+12+18=88$. There are $n = 8$ numbers. Mean $\bar{x}=\frac{88}{8}=11$

Step 14: Calculate median for fourth list

Arrange in ascending - order: $4,6,8,10,12,14,16,18$. Median $=\frac{10 + 12}{2}=11$

Step 15: Calculate standard deviation for fourth list

\[ \begin{align*} \sum_{i = 1}^{n}(x_i-\bar{x})^2&=(4 - 11)^2+(6 - 11)^2+(8 - 11)^2+(10 - 11)^2+(12 - 11)^2+(14 - 11)^2+(16 - 11)^2+(18 - 11)^2\\ &=49+25+9+1+1+9+25+49\\ &=178 \end{align*} \]
Variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}=\frac{178}{8}=22.25$. Standard deviation $s=\sqrt{22.25}\approx4.72$

Step 16: Calculate range for fourth list

Maximum value $ = 18$, minimum value $ = 4$. Range $=18 - 4 = 14$

Answer:

**First list**:
Mean $\approx4.18$, Median $ = 4$, Standard deviation $\approx2.69$, Range $ = 8$
**Second list**:
Mean $ = 18$, Median $ = 15.5$, Standard deviation $\approx7.14$, Range $ = 22$
**Third list**:
Mean $ = 2822$, Median $ = 2139$, Standard deviation $\approx1750.60$, Range $ = 5120$
**Fourth list**:
Mean $ = 11$, Median $ = 11$, Standard deviation $\approx4.72$, Range $ = 14$

Answer:

First list:
Mean $\approx4.18$, Median $ = 4$, Standard deviation $\approx2.69$, Range $ = 8$
Second list:
Mean $ = 18$, Median $ = 15.5$, Standard deviation $\approx7.14$, Range $ = 22$
Third list:
Mean $ = 2822$, Median $ = 2139$, Standard deviation $\approx1750.60$, Range $ = 5120$
Fourth list:
Mean $ = 11$, Median $ = 11$, Standard deviation $\approx4.72$, Range $ = 14$