Question
What is the standard deviation of the set of numbers below (round to the nearest tenth)? 82, 87, 90, 91, 68, 73, 55, 92, 100, 88, 73, 32, 68
Answers
GPT-4o mini
To calculate the standard deviation of the set of numbers, we can follow these steps:
1. **Calculate the mean (average)**.
2. **Calculate the squared differences from the mean**.
3. **Calculate the variance (average of squared differences)**.
4. **Calculate the standard deviation (square root of variance)**.
Let's begin:
### Step 1: Calculate the mean
First, we add all the numbers together:
\[
82 + 87 + 90 + 91 + 68 + 73 + 55 + 92 + 100 + 88 + 73 + 32 + 68 = 999
\]
Now, we divide by the number of values (13):
\[
\text{Mean} = \frac{999}{13} \approx 76.0769
\]
### Step 2: Calculate the squared differences from the mean
Next, we subtract the mean from each number, square the result, and then sum those squared differences:
\[
\begin{align*}
(82 - 76.0769)^2 & \approx (5.9231)^2 \approx 35.0291 \\
(87 - 76.0769)^2 & \approx (10.9231)^2 \approx 119.3585 \\
(90 - 76.0769)^2 & \approx (13.9231)^2 \approx 193.0561 \\
(91 - 76.0769)^2 & \approx (14.9231)^2 \approx 222.8172 \\
(68 - 76.0769)^2 & \approx (-8.0769)^2 \approx 65.0836 \\
(73 - 76.0769)^2 & \approx (-3.0769)^2 \approx 9.4661 \\
(55 - 76.0769)^2 & \approx (-21.0769)^2 \approx 444.2497 \\
(92 - 76.0769)^2 & \approx (15.9231)^2 \approx 253.7402 \\
(100 - 76.0769)^2 & \approx (23.9231)^2 \approx 571.0375 \\
(88 - 76.0769)^2 & \approx (11.9231)^2 \approx 142.1716 \\
(73 - 76.0769)^2 & \approx (-3.0769)^2 \approx 9.4661 \\
(32 - 76.0769)^2 & \approx (-44.0769)^2 \approx 1944.5044 \\
(68 - 76.0769)^2 & \approx (-8.0769)^2 \approx 65.0836 \\
\end{align*}
\]
Now we sum these squared differences:
\[
35.0291 + 119.3585 + 193.0561 + 222.8172 + 65.0836 + 9.4661 + 444.2497 + 253.7402 + 571.0375 + 142.1716 + 9.4661 + 1944.5044 + 65.0836 \approx 3194.4602
\]
### Step 3: Calculate the variance
Next, we calculate the variance. Since we have a sample (not a full population) we divide by \( n - 1 = 12 \):
\[
\text{Variance} = \frac{3194.4602}{12} \approx 266.205
\]
### Step 4: Calculate the standard deviation
Finally, we take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} \approx \sqrt{266.205} \approx 16.3
\]
So the standard deviation of the set of numbers, rounded to the nearest tenth, is approximately **16.3**.
1. **Calculate the mean (average)**.
2. **Calculate the squared differences from the mean**.
3. **Calculate the variance (average of squared differences)**.
4. **Calculate the standard deviation (square root of variance)**.
Let's begin:
### Step 1: Calculate the mean
First, we add all the numbers together:
\[
82 + 87 + 90 + 91 + 68 + 73 + 55 + 92 + 100 + 88 + 73 + 32 + 68 = 999
\]
Now, we divide by the number of values (13):
\[
\text{Mean} = \frac{999}{13} \approx 76.0769
\]
### Step 2: Calculate the squared differences from the mean
Next, we subtract the mean from each number, square the result, and then sum those squared differences:
\[
\begin{align*}
(82 - 76.0769)^2 & \approx (5.9231)^2 \approx 35.0291 \\
(87 - 76.0769)^2 & \approx (10.9231)^2 \approx 119.3585 \\
(90 - 76.0769)^2 & \approx (13.9231)^2 \approx 193.0561 \\
(91 - 76.0769)^2 & \approx (14.9231)^2 \approx 222.8172 \\
(68 - 76.0769)^2 & \approx (-8.0769)^2 \approx 65.0836 \\
(73 - 76.0769)^2 & \approx (-3.0769)^2 \approx 9.4661 \\
(55 - 76.0769)^2 & \approx (-21.0769)^2 \approx 444.2497 \\
(92 - 76.0769)^2 & \approx (15.9231)^2 \approx 253.7402 \\
(100 - 76.0769)^2 & \approx (23.9231)^2 \approx 571.0375 \\
(88 - 76.0769)^2 & \approx (11.9231)^2 \approx 142.1716 \\
(73 - 76.0769)^2 & \approx (-3.0769)^2 \approx 9.4661 \\
(32 - 76.0769)^2 & \approx (-44.0769)^2 \approx 1944.5044 \\
(68 - 76.0769)^2 & \approx (-8.0769)^2 \approx 65.0836 \\
\end{align*}
\]
Now we sum these squared differences:
\[
35.0291 + 119.3585 + 193.0561 + 222.8172 + 65.0836 + 9.4661 + 444.2497 + 253.7402 + 571.0375 + 142.1716 + 9.4661 + 1944.5044 + 65.0836 \approx 3194.4602
\]
### Step 3: Calculate the variance
Next, we calculate the variance. Since we have a sample (not a full population) we divide by \( n - 1 = 12 \):
\[
\text{Variance} = \frac{3194.4602}{12} \approx 266.205
\]
### Step 4: Calculate the standard deviation
Finally, we take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} \approx \sqrt{266.205} \approx 16.3
\]
So the standard deviation of the set of numbers, rounded to the nearest tenth, is approximately **16.3**.