What is the Mean?

The Mean is one of the most fundamental and important concepts in statistics. It represents the central tendency of data and is frequently used in everyday life.

Question: How do you calculate the mean?

The mean is the sum of all observations divided by the number of observations.

Mathematically:

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$

Where:

• $\bar{x}$ (x-bar) represents the mean
• $n$ is the number of observations
• $x_i$ is each individual observation

Example: Average Test Scores

Let's say 7 students received the following math test scores:

23, 45, 67, 89, 12, 34, 56

Calculate the mean:

(2 decimal places)확인

Properties of the Mean

The mean is the point where the sum of distances from all observations is minimized.

Key properties of the mean:

• Balance point: The mean acts as the center of gravity of the data
• Sensitivity: It is heavily influenced by extreme values (outliers)
• Uses all values: Every data point contributes to the mean calculation

Effect of Outliers

Let's examine how outliers affect the mean.

You can see that a single outlier can significantly change the mean!

Practice Problem

Calculate the mean of the following data and check for outliers:

15, 18, 21, 19, 17, 20, 16, 95

Calculate the mean:

(2 decimal places)확인

95 is a clear outlier, so the mean alone may not be a good representative of this data.

In such cases:

• Consider using the median together with the mean
• Remove outliers before calculating the mean
• Use the trimmed mean

We'll learn about the median in the next lesson!

Key Takeaways

• The mean is the sum of all values divided by the count
• It's the most basic measure of central tendency
• It's sensitive to outliers
• In Python, use np.mean(), in R, use mean()