To find the mean, median, and mode of the data set: 1, 36, 42, 40, 39, and 42, we can follow these steps:
Data Set
- Data: 1, 36, 42, 40, 39, 42
Mean
The mean (average) is calculated by summing all the values and dividing by the number of values.
\[ \text{Mean} = \frac{(1 + 36 + 42 + 40 + 39 + 42)}{6} = \frac{200}{6} \approx 33.33 \]
Median
The median is the middle value when the numbers are arranged in ascending order. In this case, we first sort the data set:
1, 36, 39, 40, 42, 42
Since there is an even number of observations (6 values), the median will be the average of the two middle numbers (39 and 40):
\[ \text{Median} = \frac{39 + 40}{2} = \frac{79}{2} = 39.5 \]
Mode
The mode is the number that appears most frequently in the data set. Here, the number 42 appears twice, while all other numbers appear once.
\[ \text{Mode} = 42 \]
Summary of Measures
- Mean: 33.33
- Median: 39.5
- Mode: 42
Which Measure Best Describes the Data?
Choosing which measure of central tendency best describes the data depends on the characteristics of the data set:
- Mean: It can be affected by extreme values (outliers). In this case, the presence of "1" significantly lowers the mean.
- Median: This measure is not as affected by outliers and gives a better sense of the center of the data.
- Mode: Indicates the most common value but does not convey information about the rest of the data.
Given the presence of the outlier (1), the median (39.5) is likely the best measure to describe the central tendency of this data set, as it provides a reasonable representation of the middle of the data without being skewed by the extreme value.