Let's work through the problem step by step to find the mean, median, mode, and range of the given set of numbers: [tex]\(\{14,68,38,65,36,57,65\}\)[/tex].
### Step 1: Arrange the Numbers in Ascending Order
First, we need to sort the numbers in ascending order:
[tex]\[ 14, 36, 38, 57, 65, 65, 68 \][/tex]
### Step 2: Calculate the Mean
The mean (average) is calculated as the sum of the numbers divided by the number of elements.
[tex]\[ \text{Mean} = \frac{\sum \text{numbers}}{n} \][/tex]
Sum of the numbers:
[tex]\[ 14 + 36 + 38 + 57 + 65 + 65 + 68 = 343 \][/tex]
Number of elements ([tex]\(n\)[/tex]):
[tex]\[ n = 7 \][/tex]
[tex]\[ \text{Mean} = \frac{343}{7} = 49 \][/tex]
### Step 3: Calculate the Median
The median is the middle number in a sorted list of numbers. Since there are 7 numbers, the median is the 4th number in the sorted list.
[tex]\[ \text{Sorted List} = 14, 36, 38, 57, 65, 65, 68 \][/tex]
[tex]\[ \text{Median} = 57 \][/tex]
### Step 4: Calculate the Mode
The mode is the number that appears most frequently in the list. From the sorted list:
[tex]\[ 14, 36, 38, 57, 65, 65, 68 \][/tex]
The number 65 appears twice, while all other numbers appear only once. Hence, the mode is:
[tex]\[ \text{Mode} = 65 \][/tex]
### Step 5: Calculate the Range
The range is the difference between the largest and smallest numbers in the set.
[tex]\[ \text{Range} = \text{Maximum} - \text{Minimum} \][/tex]
[tex]\[ \text{Maximum} = 68 \][/tex]
[tex]\[ \text{Minimum} = 14 \][/tex]
[tex]\[ \text{Range} = 68 - 14 = 54 \][/tex]
### Summary
- Mean: 49
- Median: 57
- Mode: 65
- Range: 54
I hope this helps you understand how to calculate the mean, median, mode, and range for this set of numbers!
