Answer :
3. Let's address the question of finding the mean, median, mode, and range for the given data set:
Data Set:
[tex]\[90, 94, 53, 68, 79, 94, 53, 65, 87, 90, 70, 69, 65, 89, 85, 53, 47, 61, 27, 80\][/tex]
Step-by-Step Solution:
a. Mean, Median, Mode, and Range
Mean:
The mean (or average) is the sum of all the numbers divided by the count of the numbers.
To find the mean of the data set:
[tex]\[ \text{Mean} = \frac{90 + 94 + 53 + 68 + 79 + 94 + 53 + 65 + 87 + 90 + 70 + 69 + 65 + 89 + 85 + 53 + 47 + 61 + 27 + 80}{20} \][/tex]
[tex]\[ \text{Mean} = \frac{1419}{20} \][/tex]
[tex]\[ \text{Mean} = 70.95 \][/tex]
Median:
The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.
First, we sort the data:
[tex]\[ 27, 47, 53, 53, 53, 61, 65, 65, 68, 69, 70, 79, 80, 85, 87, 89, 90, 90, 94, 94 \][/tex]
There are 20 numbers in the data set (an even number), so the median is the average of the 10th and 11th numbers:
[tex]\[ \text{Median} = \frac{69 + 70}{2} \][/tex]
[tex]\[ \text{Median} = \frac{139}{2} \][/tex]
[tex]\[ \text{Median} = 69.5 \][/tex]
Mode:
The mode is the number that appears most frequently in the data set.
From our sorted data, we can see that 53 appears three times, more frequently than any other number. Therefore:
[tex]\[ \text{Mode} = 53 \][/tex]
Range:
The range is the difference between the maximum and minimum values in the data set.
[tex]\[ \text{Range} = 94 - 27 \][/tex]
[tex]\[ \text{Range} = 67 \][/tex]
To summarize, we have:
[tex]\[ \text{Mean} = 70.95 \][/tex]
[tex]\[ \text{Median} = 69.5 \][/tex]
[tex]\[ \text{Mode} = 53 \][/tex]
[tex]\[ \text{Range} = 67 \][/tex]
b. How do you find the mean?
To find the mean, sum all the numbers in the data set and then divide the sum by the total count of numbers:
[tex]\[ \text{Mean} = \frac{\sum \text{of all values}}{\text{number of values}} \][/tex]
c. How do I calculate the median?
To calculate the median:
1. Arrange the data in ascending order.
2. Find the middle number. If the count of numbers is odd, the median is the middle number. If the count is even, the median is the average of the two middle numbers.
d. How do I calculate mode?
The mode is the value that appears most frequently in the data set. Identify the mode by counting the frequency of each value. The value with the highest frequency is the mode.
e. What is the relation between mean, mode, and median?
In a symmetrical distribution, the mean, median, and mode are all equal. In a skewed distribution, they differ:
- For a positively skewed distribution: Mode < Median < Mean
- For a negatively skewed distribution: Mean < Median < Mode
5. Pictograph:
To draw a pictograph, you'd need a proper scale. Let's use the favorite beverages data:
| Favorite Beverages | Tea | Coffee | Green Tea | Iced Tea | Cold Coffee |
|--------------------|-----|--------|-----------|----------|-------------|
| Number of People | 21 | 18 | 3 | 8 | 12 |
Choose a scale:
Let's say each pictorial representation (e.g., a cup) represents 3 people.
Then the pictograph would be:
- Tea: ☕☕☕☕☕☕☕ (7 cups, 21 people)
- Coffee: ☕☕☕☕☕☕ (6 cups, 18 people)
- Green Tea: ☕ (1 cup, 3 people)
- Iced Tea: ☕☕☕ (3 cups, 9 people)
- Cold Coffee: ☕☕☕☕ (4 cups, 12 people)
You can draw each cup or symbol representing 3 people on your graph paper to represent the data visually in a meaningful way.
Data Set:
[tex]\[90, 94, 53, 68, 79, 94, 53, 65, 87, 90, 70, 69, 65, 89, 85, 53, 47, 61, 27, 80\][/tex]
Step-by-Step Solution:
a. Mean, Median, Mode, and Range
Mean:
The mean (or average) is the sum of all the numbers divided by the count of the numbers.
To find the mean of the data set:
[tex]\[ \text{Mean} = \frac{90 + 94 + 53 + 68 + 79 + 94 + 53 + 65 + 87 + 90 + 70 + 69 + 65 + 89 + 85 + 53 + 47 + 61 + 27 + 80}{20} \][/tex]
[tex]\[ \text{Mean} = \frac{1419}{20} \][/tex]
[tex]\[ \text{Mean} = 70.95 \][/tex]
Median:
The median is the middle value when the numbers are arranged in ascending order. If there is an even number of observations, the median is the average of the two middle numbers.
First, we sort the data:
[tex]\[ 27, 47, 53, 53, 53, 61, 65, 65, 68, 69, 70, 79, 80, 85, 87, 89, 90, 90, 94, 94 \][/tex]
There are 20 numbers in the data set (an even number), so the median is the average of the 10th and 11th numbers:
[tex]\[ \text{Median} = \frac{69 + 70}{2} \][/tex]
[tex]\[ \text{Median} = \frac{139}{2} \][/tex]
[tex]\[ \text{Median} = 69.5 \][/tex]
Mode:
The mode is the number that appears most frequently in the data set.
From our sorted data, we can see that 53 appears three times, more frequently than any other number. Therefore:
[tex]\[ \text{Mode} = 53 \][/tex]
Range:
The range is the difference between the maximum and minimum values in the data set.
[tex]\[ \text{Range} = 94 - 27 \][/tex]
[tex]\[ \text{Range} = 67 \][/tex]
To summarize, we have:
[tex]\[ \text{Mean} = 70.95 \][/tex]
[tex]\[ \text{Median} = 69.5 \][/tex]
[tex]\[ \text{Mode} = 53 \][/tex]
[tex]\[ \text{Range} = 67 \][/tex]
b. How do you find the mean?
To find the mean, sum all the numbers in the data set and then divide the sum by the total count of numbers:
[tex]\[ \text{Mean} = \frac{\sum \text{of all values}}{\text{number of values}} \][/tex]
c. How do I calculate the median?
To calculate the median:
1. Arrange the data in ascending order.
2. Find the middle number. If the count of numbers is odd, the median is the middle number. If the count is even, the median is the average of the two middle numbers.
d. How do I calculate mode?
The mode is the value that appears most frequently in the data set. Identify the mode by counting the frequency of each value. The value with the highest frequency is the mode.
e. What is the relation between mean, mode, and median?
In a symmetrical distribution, the mean, median, and mode are all equal. In a skewed distribution, they differ:
- For a positively skewed distribution: Mode < Median < Mean
- For a negatively skewed distribution: Mean < Median < Mode
5. Pictograph:
To draw a pictograph, you'd need a proper scale. Let's use the favorite beverages data:
| Favorite Beverages | Tea | Coffee | Green Tea | Iced Tea | Cold Coffee |
|--------------------|-----|--------|-----------|----------|-------------|
| Number of People | 21 | 18 | 3 | 8 | 12 |
Choose a scale:
Let's say each pictorial representation (e.g., a cup) represents 3 people.
Then the pictograph would be:
- Tea: ☕☕☕☕☕☕☕ (7 cups, 21 people)
- Coffee: ☕☕☕☕☕☕ (6 cups, 18 people)
- Green Tea: ☕ (1 cup, 3 people)
- Iced Tea: ☕☕☕ (3 cups, 9 people)
- Cold Coffee: ☕☕☕☕ (4 cups, 12 people)
You can draw each cup or symbol representing 3 people on your graph paper to represent the data visually in a meaningful way.
