MYP Extended Mathematics 8.2 Standard Deviation Calculation Notes

Sigma Notation And Summary Statistics Make Calculations Efficient

In statistics we often use sigma notation to write sums compactly.

1. Two summary statistics appear frequently:
   1. $\sum x$: the sum of all data values
   2. $\sum x^2$: the sum of the squares of the data values
2. Many calculators (GDCs) can provide these directly on a summary statistics screen, which is helpful when the dataset is large.

Mean And Mean Of Squares Are Not The Same

1. The mean of a dataset is $$\mu = \frac{\sum x}{n}$$ where $n$ is the number of values.
2. Students sometimes assume that squaring the mean is the same as the mean of the squared values, but they are generally different:
   1. Square of the mean: $\mu^2 = \left(\frac{\sum x}{n}\right)^2$
   2. Mean of squares: $\frac{\sum x^2}{n}$
3. This distinction matters because the efficient standard deviation formula uses both $\frac{\sum x^2}{n}$ and $\left(\frac{\sum x}{n}\right)^2$.

Population Standard Deviation From Raw Data

If your data represents the entire population you care about (not just a sample), use the population standard deviation.

Step-By-Step Method (Raw Data)

1. Find the mean $\mu$.
2. Compute each deviation $(x-\mu)$.
3. Square each deviation $(x-\mu)^2$.
4. Add the squared deviations, then divide by $n$.
5. Take the square root.

A Faster Population Formula Using Σx And Σx²

1. Starting from $$\sigma = \sqrt{\frac{\sum (x-\mu)^2}{n}},$$ expanding and simplifying leads to a very practical form: $$\sigma = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2}$$
2. This formula shows that to compute $\sigma$ you only need:
   1. $n$
   2. $\sum x$
   3. $\sum x^2$

Standard Deviation For Frequency Tables (Discrete Data)

1. Sometimes data are given as a frequency table, where each value $x$ occurs $f$ times.
2. The mean becomes $$\mu = \frac{\sum fx}{n}$$ where $n=\sum f$.
3. The population standard deviation for a discrete frequency table is $$\sigma = \sqrt{\frac{\sum f(x-\mu)^2}{n}}$$
4. This is the same idea as before, but each squared deviation is counted according to how often the value occurs.

Sample Standard Deviation Estimates A Population Spread

1. Often you only have a sample from a larger population (for example, 10 days of bus lateness to estimate lateness for the whole town).
2. In that case, you typically use a slightly different denominator.

1. The key change is dividing by $n-1$ instead of $n$.
2. This adjustment helps make the estimate less biased when using a sample.

Combining Groups: How The Mean Updates

1. When two populations (or groups) are combined, you can find the combined mean without listing every value again:
2. If group 1 has $n_1$ values with sum $\sum x_1$ and group 2 has $n_2$ values with sum $\sum x_2$, then the combined mean is $$\mu{1+2} = \frac{\sum x_1 + \sum x_2}{n_1 + n_2}$$
3. This idea is an example of why sigma notation is powerful: you track totals and counts.