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Understanding how to calculate the sample mean is fundamental in statistics. The sample mean, often referred to as the average, represents the central point of a data set. To find it, you simply add together all the values in your sample and then divide by the number of values.

For the class representative popularity ratings, we have five values which are: 3, 2, 0, 9, and 1. By adding these up, we get a total of 15. Since there are five values, we divide 15 by 5 to get a sample mean of 3. This suggests that, on average, the popularity rating is 3 on a scale from 0 to 10.

A correct calculation of the sample mean provides a solid foundation for further statistical analysis and gives us an initial idea of where our data is centered.

The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range.

The calculation involves several steps, beginning with finding the mean, as previously described. Next, we calculate the deviation of each value from the mean, which represents how far each value is from the mean. We then square these deviations to ensure they are positive, and sum them up. This sum is then divided by the number of values minus one to account for sample variance, which in our case gives us 12.75.

The final step is to take the square root of this result to return the units to their original, giving us a standard deviation of approximately 3.57. This tells us about the variability of our popularity ratings.

Variance is another measure of dispersion in your data, similar to standard deviation but with a slight difference, as it is the average squared deviation from the mean. To compute the variance, follow the previous step and calculate each data point's deviation from the mean, square these deviations to get rid of any negative signs, and then average these squared numbers.

In our example, the sum of the squared deviations is 51. We divide this sum by the number of data points minus one (N-1), which is called the 'degrees of freedom.' This adjustment helps to correct the bias in the estimation of the population variance from a sample. The variance of our popularity ratings is 12.75, which offers significant insights when comparing datasets or understanding the data's spread.

After you've computed the standard deviation and variance, you can analyze the deviations of data points from the mean to understand the distribution. This is where the Empirical Rule comes into play, which states that for a normal distribution, nearly 68% of the data will fall within plus or minus one standard deviation from the mean.

In our exercise, with a mean of 3 and a standard deviation of approximately 3.57, we would normally expect that most ratings would be between -0.57 and 6.57. However, since scores can't be negative, the practical range starts from 0. Therefore, we can say that around 68% of the scores from the class will be between 0 and 6.57, giving a more concrete understanding of how the ratings for the class representative are distributed. This empirical analysis is crucial when making predictions or hypotheses about our data's behavior.