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math: add central tendency and dispersion

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@ -150,6 +150,62 @@ A **box-and-whisker plot** is a visual representation of the **"5-number summary
### Measures of central tendency
The **mean** is the sum of all values divided by the total number of values. $\bar{x}$ represents the mean of a sample while $µ$ represents the mean of a population.
$$\bar{x}=\frac{\Sigma x}{n}$$ where $n$ is equal to the number of values in the data set.
In grouped data, the mean can only be estimated, and is equal to the average of the sum of midpoint of all classes multiplied by their class frequency.
$$\bar{x} = \frac{\Sigma x_i f_i}{\Sigma f_i}$$ where $x_i$ is the midpoint of the $i$th class and $f_i$ is the frequency of the $i$th class.
The **median** is the middle value when the data set is sorted. If the data set has an even number of values, the median is the mean of the two centre-most values.
In grouped data, the median class is the class of the median value.
The **mode** is the value that appears most often.
!!! definition
- **Unimodal**: A data set with one mode.
- **Bimodal**: A data set with two modes.
- **Multimodal**: A data set with more than two modes.
- **No mode**: A data set with no values occurring more than once.
In grouped data, the **modal class** is the class with the greatest frequency.
### Measures of dispersion
These are used to quantify the variability or spread of the data set.
The **range** of a data set is simple to calculate but is easily thrown off by outliers.
$$R = \max - \min$$
The **variance** ($\sigma^2$) and **standard deviation** ($\sigma$) of a data set are more useful. The standard deviation indicates how closely the values of a data set are clustered around the mean.
$$\sigma = \sqrt{\frac{\Sigma f_i (x_i - \bar{x})^2}{n}}$$ where $f_i$ is the frequency of the $i$th class, $x_i$ is the midpoint of the $i$th class, $\bar{x}$ is the mean of the whole data set, and $n$ is the number of values in the data set.
For ungrouped data, assume $f_i = 1$.
In a typical bell-shaped distribution:
- 68% of data lie within 1 standard deviation of the mean ($\bar{x} ± \sigma$)
- 95% of data lie within 2 standard deviations of the mean ($\bar{x} ± 2\sigma$)
- 99.7% of data lie within 3 standard deviations of the mean ($\bar{x} ± 3\sigma$)
- any data outside 3 standard deviations of the mean can be considered outliers
!!! info
The **points of inflection** (when the curve changes direction) of a normal bell curve occur at $\bar{x} ± \sigma$.
### Constant operations on mean and standard deviation
When adding or subtracting a constant to a whole data set, the mean is increased/decreased by that constant, but the standard deviation **does not change**.
When multiplying or dividing a constant to a whole data set, both the mean and the standard deviation are multiplied or divided by that constant, respectively.
### Linear correlation and regression
**Correlation does not imply causation**.
## Resources
- [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)