The mean, median, and mode are measures of central tendency, each providing different insights into the distribution of a dataset. Here’s how they differ:

**1. Mean**

**Definition:**The mean is the average of all the data points in a dataset. It is calculated by summing all the values and then dividing by the total number of values.

**Formula:**

$\text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n}$

where $x_i$ are the data points and $n$ is the number of data points.

**Sensitivity:**The mean is sensitive to outliers, meaning that extremely high or low values can significantly affect it.

**2. Median**

**Definition:**The median is the middle value of a dataset when it is ordered from smallest to largest. If the dataset has an even number of observations, the median is the average of the two middle values.

**Calculation:**

- For an odd number of data points: the median is the value at the center.For an even number of data points: the median is the average of the two central values.

**Sensitivity:**The median is less sensitive to outliers compared to the mean, making it a better measure of central tendency for skewed distributions.

**3. Mode**

**Definition:**The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all if all values are unique.

**Sensitivity:**The mode is not affected by outliers, but it may not provide a meaningful measure of central tendency in some datasets, especially those with a uniform distribution or multiple modes.

**Summary of Differences:**

**Mean**is useful for datasets without extreme outliers and where you want the average value.

**Median**is preferred for skewed distributions or when you want a measure that is not affected by outliers.

**Mode**is used when the most common value is of interest, particularly in categorical data.

**Example:**

Consider the dataset: [3, 7, 8, 8, 10, 12, 15].

**Mean:**$\frac{3 + 7 + 8 + 8 + 10 + 12 + 15}{7} = 9$**Median:**Since the number of data points is odd, the median is the 4th value, which is 8.**Mode:**The value 8 appears most frequently, so the mode is 8.

In this example, the mean is 9, the median is 8, and the mode is 8. The mean is slightly higher due to the influence of the larger values (12 and 15).

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