Question: scatter graphs, standard deviation, mean, median, mode, percentile and quartile formulas
1. **Problem Statement:**
Explain the formulas for scatter graphs, standard deviation, mean, median, mode, percentile, and quartile.
2. **Scatter Graphs:**
A scatter graph plots points $(x_i, y_i)$ to show the relationship between two variables.
3. **Mean (Average):**
The mean of $n$ numbers $x_1, x_2, ..., x_n$ is given by:
$$\text{Mean} = \frac{\sum_{i=1}^n x_i}{n}$$
It represents the central value.
4. **Median:**
The median is the middle value when data is ordered. If $n$ is odd, median is the middle number. If even, median is the average of the two middle numbers.
5. **Mode:**
The mode is the value that appears most frequently in the data set.
6. **Standard Deviation:**
Measures spread of data around the mean. For data $x_1, x_2, ..., x_n$ with mean $\bar{x}$:
$$\text{Standard Deviation} = \sqrt{\frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2}$$
7. **Percentile:**
The $k$th percentile is the value below which $k\%$ of data falls.
8. **Quartiles:**
Divide data into four equal parts:
- $Q_1$ (first quartile) is the 25th percentile.
- $Q_2$ (second quartile) is the median (50th percentile).
- $Q_3$ (third quartile) is the 75th percentile.
These formulas help summarize and analyze data distributions clearly.