Question

159,175,189,190,191,202,208,210,218,220,232,234,234,253,253,261,265,275,295,295,304,305,308,308 mean 241 calculate the variance of the data

Answer 1

To calculate the variance of the given data, we need to follow a few steps. We'll first find the mean, but since you provided the mean (241), we will use it in our variance calculation.

### Steps to Calculate Variance:

1. **List of Data Points**: We have the data points as follows:
\[
159, 175, 189, 190, 191, 202, 208, 210, 218, 220, 232, 234, 234, 253, 253, 261, 265, 275, 295, 295, 304, 305, 308, 308
\]

2. **Count the Number of Data Points (n)**:
There are 24 data points.

3. **Calculate the Squared Differences from the Mean**:
We will calculate \((x_i - \text{mean})^2\) for each data point \(x_i\).

4. **Sum of Squared Differences**:
We will sum up all the squared differences.

5. **Calculate Variance**:
The variance is given by:
\[
\text{Variance} = \frac{\text{Sum of Squared Differences}}{n}
\]

### Calculating the Squared Differences:

\[
\begin{align*}
(159 - 241)^2 & = ( -82 )^2 = 6724 \\
(175 - 241)^2 & = ( -66 )^2 = 4356 \\
(189 - 241)^2 & = ( -52 )^2 = 2704 \\
(190 - 241)^2 & = ( -51 )^2 = 2601 \\
(191 - 241)^2 & = ( -50 )^2 = 2500 \\
(202 - 241)^2 & = ( -39 )^2 = 1521 \\
(208 - 241)^2 & = ( -33 )^2 = 1089 \\
(210 - 241)^2 & = ( -31 )^2 = 961 \\
(218 - 241)^2 & = ( -23 )^2 = 529 \\
(220 - 241)^2 & = ( -21 )^2 = 441 \\
(232 - 241)^2 & = ( -9 )^2 = 81 \\
(234 - 241)^2 & = ( -7 )^2 = 49 \\
(234 - 241)^2 & = ( -7 )^2 = 49 \\
(253 - 241)^2 & = ( 12 )^2 = 144 \\
(253 - 241)^2 & = ( 12 )^2 = 144 \\
(261 - 241)^2 & = ( 20 )^2 = 400 \\
(265 - 241)^2 & = ( 24 )^2 = 576 \\
(275 - 241)^2 & = ( 34 )^2 = 1156 \\
(295 - 241)^2 & = ( 54 )^2 = 2916 \\
(295 - 241)^2 & = ( 54 )^2 = 2916 \\
(304 - 241)^2 & = ( 63 )^2 = 3969 \\
(305 - 241)^2 & = ( 64 )^2 = 4096 \\
(308 - 241)^2 & = ( 67 )^2 = 4489 \\
(308 - 241)^2 & = ( 67 )^2 = 4489 \\
\end{align*}
\]

### Sum of Squared Differences:
Now let's sum these squared differences:
\[
6724 + 4356 + 2704 + 2601 + 2500 + 1521 + 1089 + 961 + 529 + 441 + 81 + 49 + 49 + 144 + 144 + 400 + 576 + 1156 + 2916 + 2916 + 3969 + 4096 + 4489 + 4489 = 49607
\]

### Calculate Variance:
Now we can calculate the variance.
\[
\text{Variance} = \frac{49607}{24} \approx 2067.79
\]

Thus, the variance of the data is approximately **2067.79**.