For the given data set (13, 16, 12, 14, 19, 12, 14, 13, 14), the exact mean is 127/9, the median is 14, and the mode is 14.
To accurately determine these statistical measures, we first arrange the data points in ascending order:
12, 12, 13, 13, 14, 14, 14, 16, 19
There are 9 data points in total.
Calculating the Mean
The mean, often referred to as the average, is calculated by summing all the values in a data set and then dividing by the total count of values.
- Sum of values: 12 + 12 + 13 + 13 + 14 + 14 + 14 + 16 + 19 = 127
- Number of values (n): 9
- Mean Calculation: Sum / n = 127 / 9
The exact mean is 127/9, which can be expressed as a repeating decimal approximately 14.11 (14.111...).
Determining the Median
The median represents the middle value in a data set when the values are arranged in ascending or descending order. If the data set contains an odd number of values, the median is the single middle value. If it contains an even number, the median is the average of the two central values.
- Ordered data: 12, 12, 13, 13, 14, 14, 14, 16, 19
- Position of median: For a data set with 9 points, the median is the ((9 + 1) / 2) = 5th value.
- Median value: Counting five positions from the beginning of the ordered list, the 5th value is 14.
Identifying the Mode
The mode is the value that appears most frequently within a data set. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode if all values occur with the same frequency.
- Count occurrences of each value in the ordered data:
- 12 appears 2 times
- 13 appears 2 times
- 14 appears 3 times
- 16 appears 1 time
- 19 appears 1 time
- Most frequent value: The number 14 occurs 3 times, which is more frequently than any other number in the set.
Summary of Results
Here's a concise overview of the calculated measures for the given data set:
| Measure | Value |
|---|---|
| Mean | 127/9 (≈14.11) |
| Median | 14 |
| Mode | 14 |
As shown by the calculations, both the median and the mode of this specific data set are 14.
