A sample of ten motors were tested and the following data were obtained for the
number of revolutions per minute the flywheel attached to the motors turned. Find
the mean, median, variance, and standard deviation.
215 308 225 236 272
300 254 260 232 216
Mean:
Median:
Variance:
Standard Deviation

We are given the following data set of motor revolutions per minute (RPM):

\[ 215, 308, 225, 236, 272, 300, 254, 260, 232, 216 \]

To find the **mean**, **median**, **variance**, and **standard deviation**, let's calculate each step-by-step.

### 1. **Mean**:
The mean is the sum of all values divided by the number of values:

\[
\text{Mean} = \frac{215 + 308 + 225 + 236 + 272 + 300 + 254 + 260 + 232 + 216}{10}
\]

\[
\text{Mean} = \frac{2518}{10} = 251.8
\]

### 2. **Median**:
The median is the middle value when the data is arranged in increasing order. For an even number of data points, the median is the average of the two middle numbers.

Arranging the data in increasing order:
\[  215, 216, 225, 232, 236, 254, 260, 272, 300, 308 \]

The middle two values are \(236\) and \(254\), so:

\[
\text{Median} = \frac{236 + 254}{2} = \frac{490}{2} = 245
\]

### 3. **Variance**:
Variance is the average of the squared differences from the mean. We first find the difference of each data point from the mean and then square it.

\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n}
\]

Let me calculate the squared differences and the variance.

### 4. **Standard Deviation**:
The standard deviation is the square root of the variance.

Let me calculate the variance and standard deviation.### Results:
- **Mean**: \( 251.8 \)
- **Median**: \( 245 \)
- **Variance**: \( 995.76 \)
- **Standard Deviation**: \( 31.56 \)

Would you like any further details or clarifications?

Here are 5 related questions to expand on this:
1. How does the variance change if we use a different sample size?
2. What is the relationship between variance and standard deviation?
3. How would these values change if we added more data points?
4. Can you explain how to compute the range and interquartile range for this data set?
5. How does the mean compare to the mode in this dataset?

**Tip**: Variance gives a sense of how much the data points deviate from the mean, but standard deviation puts this in the original unit of measurement, making it easier to interpret.

A sample of ten motors were tested and the following data were obtained for the number of revolutions per minute the flywheel attached to the motors turned. Find the mean, median, variance, and standard deviation.
215 308 225 236 272 300 254 260 232 216
Mean:
Median:
Variance:
Standard Deviation:

Learn how to calculate the mean, median, variance, and standard deviation for a sample of ten motor RPMs: 215, 308, 225, 236, 272, 300, 254, 260, 232, and 216. This problem covers key statistical concepts and provides a detailed step-by-step solution. Suitable for students in Grades 9-12.

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