Overview of Measures of Center (Ungrouped Data) - Mean, Median, and Mode with formulas and explanations.

The image provides an overview of the Measures of Center (Ungrouped Data), also known as Measures of Central Tendency. These measures help describe the center or middle set of data. The three primary measures discussed are:

1. Mean
2. Median
3. Mode

Explanation of Each Measure:

#### 1. Mean
- Definition: The mean is the average value of a dataset.
- Formula:
\[
\bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}
\]
or equivalently,
\[
\bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i
\]
- Explanation: To calculate the mean, sum all the values in the dataset and divide by the total number of values (\( n \)).
- Where \( n \) is: The number of cases or observations in the dataset.

#### 2. Median
- Definition: The median is the middle value in a dataset when the values are arranged in ascending or descending order.
- Formula for Finding the Median Position:
\[
\text{Median Rank} = \frac{n + 1}{2}
\]
where \( n \) is the number of cases.
- Explanation:
- If the number of elements (\( n \)) is odd, the median is the value at the position given by \( \frac{n + 1}{2} \).
- If the number of elements (\( n \)) is even, the median is the average of the two middle values. These two middle values are located at positions \( \frac{n}{2} \) and \( \frac{n}{2} + 1 \).

#### 3. Mode
- Definition: The mode is the most frequently occurring value in the dataset.
- Explanation:
- A dataset can have one mode (unimodal), two modes (bimodal), more than two modes (multimodal), or no mode at all if all values occur with equal frequency.
- If all values in the dataset occur with the same frequency, the dataset has no mode.

Summary of Key Points:

- Mean: Arithmetic average of all values.
- Median: Middle value when data is ordered.
- Mode: Most frequent value in the dataset.

Example to Illustrate:

Let's consider the dataset: \( 5, 7, 9, 10, 12 \).

1. Mean:
\[
\bar{X} = \frac{5 + 7 + 9 + 10 + 12}{5} = \frac{43}{5} = 8.6
\]

2. Median:
- The dataset is already ordered.
- Since \( n = 5 \) (odd), the median is the value at position \( \frac{5 + 1}{2} = 3 \).
- The third value is \( 9 \).
- Therefore, the median is \( 9 \).

3. Mode:
- All values occur only once.
- Therefore, there is no mode.

Final Answer:

\[
\boxed{\text{Mean, Median, Mode}}
\]