wyatt is bored and trying to find the middle of the first 20 integers. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11…

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a set of data $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here $n = 20$, and $\sum_{i=1}^{20}i=\frac{20\times(20 + 1)}{2}=210$. So $\bar{x}=\frac{210}{20}=10.5$.

Step2: Calculate the standard - deviation

The formula for the 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 $i$: $\sum_{i = 1}^{20}(x_i - 10.5)^2=(1 - 10.5)^2+(2 - 10.5)^2+\cdots+(20 - 10.5)^2$. $\sum_{i=1}^{20}(x_i - 10.5)^2=(- 9.5)^2+(-8.5)^2+\cdots+(9.5)^2$. $=9.5^2+8.5^2+\cdots+1.5^2+0.5^2+0.5^2+1.5^2+\cdots+8.5^2+9.5^2$. $=2\times(0.5^2 + 1.5^2+\cdots+9.5^2)$. $0.5^2=0.25$, $1.5^2 = 2.25$, $\cdots$, $9.5^2=90.25$. $\sum_{k = 1}^{9}(0.5 + k)^2=\sum_{k = 1}^{9}(0.25+ k + k^2)=9\times0.25+\sum_{k = 1}^{9}k+\sum_{k = 1}^{9}k^2$. $=2.25+\frac{9\times(9 + 1)}{2}+\frac{9\times(9 + 1)\times(2\times9+1)}{6}$. $=2.25 + 45+285=332.25$. $\sum_{i=1}^{20}(x_i - 10.5)^2=2\times332.25 = 664.5$. $s=\sqrt{\frac{664.5}{19}}\approx5.916$.

Step3: Find the minimum

The minimum value of the data set $1,2,\cdots,20$ is $1$.

Step4: Find the first - quartile (Q1)

The first quartile is the median of the lower half of the data. Since $n = 20$, the lower half is the first 10 numbers. The median of $1,2,\cdots,10$ is $\frac{5+6}{2}=5.5$.

Step5: Find the median

Since $n = 20$ (an even - numbered data set), the median is the average of the 10th and 11th ordered data values. The 10th value is 10 and the 11th value is 11, so the median is $\frac{10 + 11}{2}=10.5$.

Step6: Find the third - quartile (Q3)

The third quartile is the median of the upper half of the data. The upper half is the last 10 numbers. The median of $11,12,\cdots,20$ is $\frac{15+16}{2}=15.5$.

Step7: Find the maximum

The maximum value of the data set $1,2,\cdots,20$ is $20$.

Answer:

	Value
Mean	$10.5$
St. Dev	$\approx5.916$
Min	$1$
Q1	$5.5$
Median	$10.5$
Q3	$15.5$
Max	$20$