91Ó°ÊÓ

To find the mean, you simply add up all the values and then divide by the number of values you added. For the pulse rates provided for nine students, calculate the mean by summing their pulse rates and then dividing by 9. The equation for the mean (or average) is given by:

\[\text{mean}(\mu) = \frac{\sum_{i=1}^{N}x_i}{N}\text{,}\] where \(N\) represents the total number of values. Here, adding the pulse rates yields a sum of 650. Therefore, the mean pulse rate is:

\[\text{mean} = \frac{650}{9} \approx 72.22\text{.}\] This value is crucial because it represents the central tendency of the data.

Population variance measures how much the values in a population differ from the mean of the population. It gives us an idea of the overall spread of the data. To calculate it, you first compute the deviation (the difference between each value and the mean), square each deviation, and then find the average of these squared deviations.

The equation for population variance is:

\[\text{Population Variance} \(\sigma^2\) = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}\text{,}\] where \(x_i =\) individual values, and \(N =\) number of values.

In this exercise, you calculate the deviations for each pulse rate and then square them, e.g., \((76 - 72.22)^2 \approx 14.33\text{,}\...\text{.}\) Adding all the squared deviations gives about 530.26. Finally, the variance is:

\[\text{Population Variance} \(\sigma^2\) = \frac{530.2584}{9} \approx 58.92\text{.}\]

A sample standard deviation measures the spread of data points within a sample, rather than an entire population. It estimates how much individual values deviate from the sample mean. This measure is often used when it is impractical or impossible to measure the entire population.

To calculate the sample standard deviation, first, find the mean of the sample. Then, calculate each deviation from this mean, square it, and sum these squared deviations. Lastly, divide by \(n - 1\) (where \(n\) is the number of values in the sample), and then take the square root of this quotient. The formula for sample standard deviation is:

\[\text{Sample Standard Deviation} \(s\) = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}\text{,}\] where \(\bar{x}\) represents the sample mean.

For example, Sample 1 with values (76, 60, 60) has a sample mean of 65.33 and a sample standard deviation of approximately 9.28. Comparing sample standard deviations against the population standard deviation (7.67), we see whether the samples overestimate or underestimate the spread.