Mean, Median, Mode, Range Calculator

Count (n): The total number of data points in the dataset.

Sum ($\sum x_i$): The sum of all values in the dataset.

Mean: The arithmetic average of all values. It's most useful for symmetrically distributed data without extreme outliers. It represents the "balancing point" of the dataset.
Formula: $$ \text{Mean} (\bar{x}) = \frac{\sum_{i=1}^{n} x_i}{n} $$ Where $x_i$ are the individual data points and $n$ is the total number of data points.

Median: The middle value of a dataset when ordered. It's particularly useful for skewed distributions or data with outliers, as it is less affected by extreme values than the mean. It represents the typical value.
Formula:

• If $n$ is odd, Median = Middle value
• If $n$ is even, Median = Average of the two middle values

Mode: The value(s) that appear most frequently in a dataset. Useful for categorical data or to identify peaks in the distribution of numerical data. A distribution can have no mode, one mode (unimodal), two modes (bimodal), or more (multimodal).
Formula: The value(s) with the highest frequency in the dataset.

Minimum: The smallest value in the dataset.

Maximum: The largest value in the dataset.

Range: The difference between the maximum and minimum values. It gives a quick sense of the total spread but is highly sensitive to outliers.
Formula: $$ \text{Range} = \text{Maximum Value} - \text{Minimum Value} $$

Mid-Range: The average of the maximum and minimum values in the dataset.
Formula: $$ \text{Mid-Range} = \frac{\text{Maximum Value} + \text{Minimum Value}}{2} $$

Quartiles (Q1, Q2, Q3) and IQR: Quartiles divide the data into four equal parts. Q1 (25th percentile) is the median of the lower half, Q2 (50th percentile) is the median, and Q3 (75th percentile) is the median of the upper half. The Interquartile Range (IQR = Q3 - Q1) measures the spread of the middle 50% of the data, making it robust to outliers.
Formulas:

• $Q_1 = \text{Value at } 25^{th} \text{ percentile}$
• $Q_2 = \text{Median}$
• $Q_3 = \text{Value at } 75^{th} \text{ percentile}$
• $IQR = Q_3 - Q_1$

Variance and Standard Deviation: Both measure the spread or dispersion of data points around the mean. Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is more interpretable as it's in the same units as the data. A higher standard deviation indicates greater variability.
Formulas (for population): $$ \text{Variance} (\sigma^2) = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n} $$ $$ \text{Standard Deviation} (\sigma) = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}} $$ Where $x_i$ are the individual data points, $\bar{x}$ is the mean, and $n$ is the total number of data points.

Standard Error of Mean (SEM): Measures how much the sample mean is likely to vary from the population mean. It indicates the precision of the sample mean.
Formula: $$ \text{SEM} = \frac{\sigma}{\sqrt{n}} $$ Where $\sigma$ is the population standard deviation and $n$ is the sample size.

Geometric Mean: A type of mean that indicates the central tendency of a set of numbers by using the product of their values. It is often used for data that grows exponentially.
Formula: $$ \text{Geometric Mean} = \sqrt[n]{x_1 \cdot x_2 \cdot \ldots \cdot x_n} $$ Where $x_i$ are the individual data points and $n$ is the total number of data points. (Note: Only applicable for positive numbers.)

Harmonic Mean: A type of average that is useful for rates or ratios. It is the reciprocal of the arithmetic mean of the reciprocals of the data points.
Formula: $$ \text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} $$ Where $x_i$ are the individual data points and $n$ is the total number of data points. (Note: Only applicable for non-zero numbers.)

Root Mean Square (RMS): Also known as the quadratic mean, it is a statistical measure of the magnitude of a varying quantity. It is especially useful when values can be positive or negative.
Formula: $$ \text{RMS} = \sqrt{\frac{\sum_{i=1}^{n} x_i^2}{n}} $$ Where $x_i$ are the individual data points and $n$ is the total number of data points.

Multi-modal distribution: Indicates that there are multiple distinct peaks or clusters in the data. This often suggests that the dataset is composed of two or more different sub-populations or processes, each with its own central tendency. For example, a bimodal distribution of heights might indicate a mix of male and female individuals.