Measures of Central Tendency in Statistics

However, it is important to note that the mean can sometimes be misleading if the data is highly skewed or contains extreme outliers.

The formulas for the mean differ depending on whether you are analyzing a population or a sample:

Population Mean ($\mu$):

$$\mu = \frac{\sum_{i=1}^{N} x_i}{N}$$

Where $N$ is the population size.

Sample Mean ($\bar{x}$):

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

Where $n$ is the observed sample size.

The Median

The median is the middle value of a data series that separates the observations into two equal halves. Unlike the mean, the median is less affected by extreme values and can sometimes be more "telling" of what is truly expected in a dataset.

To find the median, you must first order the data from smallest to largest (or largest to smallest). You can then determine the median position using the following formula:

$$\text{Median position} = \left(\frac{n + 1}{2}\right)$$

• If the sample size ($n$) is odd: The median is the exact middle value.
• If the sample size ($n$) is even: The median is the average of the two middle values.

Worked Example: Finding the Median

Given the dataset: $[1, 3, 4, 8, 13]$

1. The data is already ordered.
2. $n = 5$ (an odd number).
3. Position $= (5 + 1) / 2 = 3$.
4. The 3rd value is 4. The median is 4.

Given the dataset: $[1, 3, 4, 8, 13, 18]$

1. $n = 6$ (an even number).
2. Position $= (6 + 1) / 2 = 3.5$.
3. We take the average of the 3rd and 4th values: $(4 + 8) / 2 = 6$. The median is 6.

The Mode

The mode is the most frequently occurring value in a data series. While it can represent an "expected" value, it is not as commonly used for that specific purpose as the mean or median.

Data Distribution and Visualizing the Center

The relationship between the mean, median, and mode can provide significant insight into the data distribution (the shape of the data).

In a perfectly symmetrical distribution, the mean, median, and mode would all be located at the same point. However, in skewed distributions, these measures pull apart. For instance, if the mean is significantly larger than the median, it suggests the data is skewed toward higher values.

Summary of Location Measures

To get a complete picture of a variable's location, statisticians often use a "5-number summary," which includes the minimum, the first quartile ($Q_1$), the median, the third quartile ($Q_3$), and the maximum.

Exam Tip: When asked to find the position of a quartile or percentile, remember that the formula gives you the position in the ordered list, not the actual value itself. Always order your data first!