Certainly! Let's break down the calculation for the mean, median, and mode of the given data.
### Data Set
```
52, 86, 76, 51, 62, 67, 70, 50,
45, 49, 54, 58, 53, 74, 64, 56,
50, 80, 10, 57, 64, 64, 43, 78,
84, 71, 55, 72, 78, 43, 42, 75,
84, 72, 69, 49, 66, 42, 65, 88
```
### 1. Mean
The mean (or average) is calculated by summing up all the values and dividing by the number of values.
Step-by-step calculation:
[tex]\[
\text{Sum} = 52 + 86 + 76 + 51 + 62 + 67 + 70 + 50 + 45 + 49 + 54 + 58 + 53 + 74 + 64 + 56 + 50 + 80 + 10 + 57 + 64 + 64 + 43 + 78 + 84 + 71 + 55 + 72 + 78 + 43 + 42 + 75 + 84 + 72 + 69 + 49 + 66 + 42 + 65 + 88
\][/tex]
Sum = 2744
We have 40 values in the dataset.
[tex]\[
\text{Mean} = \frac{\text{Sum}}{\text{Number of values}} = \frac{2744}{40} = 68.6
\][/tex]
### 2. Median
The median is the middle value in a sorted list of numbers. If the list has an even number of elements, the median is the average of the two middle numbers.
Step-by-step calculation:
First, sort the dataset:
[tex]\[
10, 42, 42, 43, 43, 45, 49, 49, 50, 50, 52, 53, 54, 55, 56, 57, 58, 62, 64, 64, 64, 66, 67, 69, 70, 71, 72, 72, 74, 75, 76, 78, 78, 80, 84, 84, 84, 86, 88
\][/tex]
There are 40 values. The two middle values are the 20th and 21st values.
Middle two values are:
64 and 64
[tex]\[
\text{Median} = \frac{64 + 64}{2} = 64
\][/tex]
### 3. Mode
The mode is the value that appears most frequently in the dataset.
Step-by-step calculation:
Identifying the most frequent value(s) in the set:
- 64 appears 3 times
- 42 appears 3 times
- 43 appears 3 times
- 84 appears 3 times
- 78 appears 2 times
Since 64, 42, 43, and 84 occur the same highest number of times (i.e., three times each), there are multiple modes.
Therefore, the modes are: 42, 43, 64, and 84.
### Result Summary
- Mean: 68.6
- Median: 64
- Mode: 42, 43, 64, 84
