Grasping the concepts of mean and standard deviation is essential for statistical analysis. This comprehensive guide explores advanced techniques for calculating these statistical measures from raw or grouped data. Focuses on the use of summation $(\Sigma x)$ and squared summation $(\Sigma x^2)$ totals, and their application in comparing two sets of data and solving complex problems. The aim is to provide a solid foundation for understanding and applying these key statistical measures.

## Mean (Average) Calculation

- Mean = Total of all values / Number of values
- Formula: $\bar{x} = \frac{\Sigma x}{n}$
**Example:**Data Set: 5, 8, 7, 10- Sum = 5 + 8 + 7 + 10 = 30
- Mean $= \frac{30}{4} = 7.5$

## Standard Deviation Calculation

- Measures how spread out data is from the mean.
- Formula: $\sigma = \sqrt{\frac{\Sigma (x - \bar{x})^2}{n}}$
- Example: Data Set: 5, 8, 7, 10
- Mean = 7.5
- Calculate each value's squared difference from the mean and sum up.
- Standard Deviation $= \sqrt{\frac{13}{4}} \approx 1.80$

## Grouped Data Calculations

- Mean: Multiply midpoints by frequency, sum up, then divide by total frequency.
- Standard Deviation: For each group, calculate squared difference from mean times frequency, sum up, then divide by total frequency.
- Example: Data: | Interval | Frequency |
- | 0-10 | 5 |
- | 10-20 | 10 |
- | 20-30 | 15 |
- Mean ≈ 18.33, Standard Deviation ≈ 7.93

## Summation and Squared Summation

- Summation $(\Sigma x)$: Add all data values.
- Squared Summation $(\Sigma x^2)$: Square each value, then sum up.
- Example: Data Set: 3, 4, 7, 9
- Summation = 23, Squared Summation = 155

## Coded Data Comparison

- Adjust data sets to a common scale.
- Calculate mean and standard deviation for comparison.
- Example: Data Set A: 5, 10, 15; Data Set B: 15, 20, 25
- Adjust B: 5, 6.67, 8.33
- Compare means and standard deviations of both sets.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.