Understandings of Descriptive Statistics

Mean:

The mean is the average of a set of numbers. It is calculated by summing up all the values in the data set and dividing by the number of values.

Equation: $\text{Mean}=\frac{{\sum }_{𝑖=1}^{𝑛}{𝑥}_{𝑖}}{𝑛}$

Use: The mean is useful when you want to find a typical value in a dataset. It's commonly used in situations where the data is evenly distributed.

Median:

The median is the middle value of a dataset when it's ordered from least to greatest. If the dataset has an even number of values, the median is the average of the two middle values.

Equation: For an odd number of values, $\text{Median}={𝑥}_{\frac{𝑛+1}{2}}$. For an even number of values, $\text{Median}=\frac{{𝑥}_{\frac{𝑛}{2}}+{𝑥}_{\frac{𝑛}{2}+1}}{2}$.

Use: The median is useful when the dataset contains outliers or extreme values, as it is not influenced by extreme values as much as the mean.

Mode:

The mode is the value that appears most frequently in a dataset. A dataset can have one mode, more than one mode (multimodal), or no mode (no value appears more than once).

Equation: No direct equation; it's the value(s) with the highest frequency.

Use: The mode is useful for categorical or discrete data, especially when you want to find the most common category or value.

MAD (Mean Absolute Deviation):

The mean absolute deviation measures the average distance between each data point and the mean of the dataset. It provides a measure of the variability in the dataset.

Equation: $\text{MAD}=\frac{{\sum }_{𝑖=1}^{𝑛}\mathrm{\mid }{𝑥}_{𝑖}-\text{Mean}\mathrm{\mid }}{𝑛}$

Use: MAD is useful for understanding the variability or dispersion of the data. It's less influenced by outliers compared to standard deviation.

Standard Deviation (σ):

The standard deviation measures the average amount of variation or dispersion of a set of values from the mean. It indicates how spread out the values in a dataset are.

Equation: $\text{SD}=\sqrt{\frac{{\sum }_{𝑖=1}^{𝑛}({𝑥}_{𝑖}-\text{Mean}{)}^2}{𝑛}}$

Use: Standard deviation is useful when you want to understand the spread of the data and how much individual values deviate from the mean. It's commonly used in normal distribution analysis.

Variance (σ²):

Variance is a measure of how much the values in a dataset differ from the mean. It is the square of the standard deviation.

Equation: $\text{Variance}=\frac{{\sum }_{𝑖=1}^{𝑛}({𝑥}_{𝑖}-\text{Mean}{)}^2}{𝑛}$

Use: Variance quantifies the dispersion of data points in the dataset. It's used along with standard deviation to understand the spread of data.

Range:

The range is the difference between the maximum and minimum values in a dataset. It provides a simple measure of the spread of data.

Equation: $\text{Range}=\text{Maximum Value}-\text{Minimum Value}$

Use: The range is useful for quickly understanding the spread of data, but it doesn't provide information about the distribution or variability within the dataset.

In summary, each statistical measure serves a different purpose and provides unique insights into the characteristics of a dataset. The choice of which measure to use depends on the type of data, the research question, and the specific characteristics of the dataset, such as the presence of outliers and the desired level of detail in the analysis.