what is range?
what is variance?
what is standard deviation using standard variance formula

### Range
The **range** in statistics is a measure of the spread or dispersion of a data set. It is the difference between the largest and the smallest value:
\[
\text{Range} = \text{Maximum value} - \text{Minimum value}
\]

### Variance
The **variance** measures how much the values in a data set differ from the mean. It is the average of the squared differences between each data point and the mean of the data set. The formulas for variance differ slightly depending on whether you're dealing with a sample or the entire population:

1. **Population Variance** (\(\sigma^2\)):
   \[
   \sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
   \]
   where:
   - \(N\) is the number of data points.
   - \(x_i\) is the \(i\)-th data point.
   - \(\mu\) is the population mean.

2. **Sample Variance** (\(s^2\)):
   \[
   s^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2
   \]
   where:
   - \(n\) is the number of sample data points.
   - \(x_i\) is the \(i\)-th sample data point.
   - \(\bar{x}\) is the sample mean.

### Standard Deviation
The **standard deviation** is the square root of the variance and provides a measure of the spread of data in the same units as the data itself. It shows how much the individual data points typically differ from the mean.

1. **Population Standard Deviation** (\(\sigma\)):
   \[
   \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}
   \]

2. **Sample Standard Deviation** (\(s\)):
   \[
   s = \sqrt{\frac{1}{n - 1} \sum_{i=1}^{n} (x_i - \bar{x})^2}
   \]

In simple terms, variance gives you an idea of the spread of the data by squaring deviations from the mean, and standard deviation translates that spread back to the original units of the data by taking the square root.

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Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How is range different from interquartile range?
2. What is the difference between variance and standard deviation?
3. Why do we divide by \(n-1\) for sample variance instead of \(n\)?
4. How do we calculate the mean if the data set includes negative values?
5. What is the practical use of variance and standard deviation in real-world scenarios?

**Tip:** For normally distributed data, around 68% of values fall within one standard deviation of the mean.

what is range? what is variance? what is standard deviation using standard variance formula

Explore key statistical concepts like range, variance, and standard deviation. This guide covers the formulas for population and sample variance, the difference between range and standard deviation, and their practical applications. Ideal for students in Grades 10-12.

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