Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure! Let's go through the step-by-step process to find the mean, median, and mode for the given frequency distribution table.
### Step 1: Calculate the Midpoints of Each Class Interval
First, we need to find the midpoints of each class interval. The midpoint ([tex]\(x_i\)[/tex]) for a class interval [tex]\((a, b)\)[/tex] is calculated using the formula:
[tex]\[ x_i = \frac{a + b}{2} \][/tex]
For the given class intervals:
- [tex]\(10-20\)[/tex]: [tex]\( \frac{10 + 20}{2} = 15.0 \)[/tex]
- [tex]\(20-30\)[/tex]: [tex]\( \frac{20 + 30}{2} = 25.0 \)[/tex]
- [tex]\(30-40\)[/tex]: [tex]\( \frac{30 + 40}{2} = 35.0 \)[/tex]
- [tex]\(40-50\)[/tex]: [tex]\( \frac{40 + 50}{2} = 45.0 \)[/tex]
- [tex]\(50-60\)[/tex]: [tex]\( \frac{50 + 60}{2} = 55.0 \)[/tex]
So, the midpoints are [tex]\(15.0, 25.0, 35.0, 45.0, 55.0\)[/tex].
### Step 2: Calculate the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is calculated using the formula:
[tex]\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
Where:
- [tex]\(x_i\)[/tex] is the midpoint of each class interval
- [tex]\(f_i\)[/tex] is the frequency of each class interval
Substitute the values:
[tex]\[ \bar{x} = \frac{(15.0 \cdot 5) + (25.0 \cdot 8) + (35.0 \cdot 12) + (45.0 \cdot 6) + (55.0 \cdot 3)}{5 + 8 + 12 + 6 + 3} \][/tex]
[tex]\[ \bar{x} = \frac{75 + 200 + 420 + 270 + 165}{34} = \frac{1130}{34} \approx 33.24 \][/tex]
### Step 3: Calculate the Median
To find the median, we need to locate the class interval containing the median. Follow these steps:
1. Calculate the cumulative frequency:
- Cumulative frequency up to [tex]\(10-20\)[/tex]: [tex]\(5\)[/tex]
- Cumulative frequency up to [tex]\(20-30\)[/tex]: [tex]\(5 + 8 = 13\)[/tex]
- Cumulative frequency up to [tex]\(30-40\)[/tex]: [tex]\(13 + 12 = 25\)[/tex]
- Cumulative frequency up to [tex]\(40-50\)[/tex]: [tex]\(25 + 6 = 31\)[/tex]
- Cumulative frequency up to [tex]\(50-60\)[/tex]: [tex]\(31 + 3 = 34\)[/tex]
2. Identify the median class, which is the class where the cumulative frequency is greater than or equal to [tex]\(\frac{N}{2}\)[/tex]:
- [tex]\(N = 34\)[/tex]; [tex]\(\frac{N}{2} = 17\)[/tex]
- The cumulative frequency first exceeds [tex]\(17\)[/tex] in the class interval [tex]\(30-40\)[/tex].
3. Calculate the median using the formula:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f_m} \right) \cdot c \][/tex]
Where:
- [tex]\(L\)[/tex] is the lower boundary of the median class: [tex]\(30\)[/tex]
- [tex]\(N\)[/tex] is the total frequency: [tex]\(34\)[/tex]
- [tex]\(F\)[/tex] is the cumulative frequency before the median class: [tex]\(13\)[/tex]
- [tex]\(f_m\)[/tex] is the frequency of the median class: [tex]\(12\)[/tex]
- [tex]\(c\)[/tex] is the class interval width: [tex]\(10\)[/tex]
Substitute the values:
[tex]\[ \text{Median} = 30 + \left( \frac{17 - 13}{12} \right) \cdot 10 = 30 + \left( \frac{4}{12} \right) \cdot 10 = 30 + 3.33 \approx 33.33 \][/tex]
### Step 4: Calculate the Mode
To find the mode, identify the modal class, which is the class interval with the highest frequency. The frequency values are [tex]\(5, 8, 12, 6, 3\)[/tex]:
- The highest frequency is [tex]\(12\)[/tex] for the class interval [tex]\(30-40\)[/tex].
Calculate the mode using the formula:
[tex]\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \cdot c \][/tex]
Where:
- [tex]\(L\)[/tex] is the lower boundary of the modal class: [tex]\(30\)[/tex]
- [tex]\(f_1\)[/tex] is the frequency of the modal class: [tex]\(12\)[/tex]
- [tex]\(f_0\)[/tex] is the frequency of the class before the modal class: [tex]\(8\)[/tex]
- [tex]\(f_2\)[/tex] is the frequency of the class after the modal class: [tex]\(6\)[/tex]
- [tex]\(c\)[/tex] is the class interval width: [tex]\(10\)[/tex]
Substitute the values:
[tex]\[ \text{Mode} = 30 + \left( \frac{12 - 8}{2 \cdot 12 - 8 - 6} \right) \cdot 10 = 30 + \left( \frac{4}{24 - 14} \right) \cdot 10 = 30 + \left( \frac{4}{10} \right) \cdot 10 = 30 + 4 = 34.00 \][/tex]
### Summary
The ordered quantities (Mean, Median, Mode) for the given frequency distribution table are:
- Mean: 33.24
- Median: 33.33
- Mode: 34.00
### Step 1: Calculate the Midpoints of Each Class Interval
First, we need to find the midpoints of each class interval. The midpoint ([tex]\(x_i\)[/tex]) for a class interval [tex]\((a, b)\)[/tex] is calculated using the formula:
[tex]\[ x_i = \frac{a + b}{2} \][/tex]
For the given class intervals:
- [tex]\(10-20\)[/tex]: [tex]\( \frac{10 + 20}{2} = 15.0 \)[/tex]
- [tex]\(20-30\)[/tex]: [tex]\( \frac{20 + 30}{2} = 25.0 \)[/tex]
- [tex]\(30-40\)[/tex]: [tex]\( \frac{30 + 40}{2} = 35.0 \)[/tex]
- [tex]\(40-50\)[/tex]: [tex]\( \frac{40 + 50}{2} = 45.0 \)[/tex]
- [tex]\(50-60\)[/tex]: [tex]\( \frac{50 + 60}{2} = 55.0 \)[/tex]
So, the midpoints are [tex]\(15.0, 25.0, 35.0, 45.0, 55.0\)[/tex].
### Step 2: Calculate the Mean
The mean ([tex]\(\bar{x}\)[/tex]) is calculated using the formula:
[tex]\[ \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \][/tex]
Where:
- [tex]\(x_i\)[/tex] is the midpoint of each class interval
- [tex]\(f_i\)[/tex] is the frequency of each class interval
Substitute the values:
[tex]\[ \bar{x} = \frac{(15.0 \cdot 5) + (25.0 \cdot 8) + (35.0 \cdot 12) + (45.0 \cdot 6) + (55.0 \cdot 3)}{5 + 8 + 12 + 6 + 3} \][/tex]
[tex]\[ \bar{x} = \frac{75 + 200 + 420 + 270 + 165}{34} = \frac{1130}{34} \approx 33.24 \][/tex]
### Step 3: Calculate the Median
To find the median, we need to locate the class interval containing the median. Follow these steps:
1. Calculate the cumulative frequency:
- Cumulative frequency up to [tex]\(10-20\)[/tex]: [tex]\(5\)[/tex]
- Cumulative frequency up to [tex]\(20-30\)[/tex]: [tex]\(5 + 8 = 13\)[/tex]
- Cumulative frequency up to [tex]\(30-40\)[/tex]: [tex]\(13 + 12 = 25\)[/tex]
- Cumulative frequency up to [tex]\(40-50\)[/tex]: [tex]\(25 + 6 = 31\)[/tex]
- Cumulative frequency up to [tex]\(50-60\)[/tex]: [tex]\(31 + 3 = 34\)[/tex]
2. Identify the median class, which is the class where the cumulative frequency is greater than or equal to [tex]\(\frac{N}{2}\)[/tex]:
- [tex]\(N = 34\)[/tex]; [tex]\(\frac{N}{2} = 17\)[/tex]
- The cumulative frequency first exceeds [tex]\(17\)[/tex] in the class interval [tex]\(30-40\)[/tex].
3. Calculate the median using the formula:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f_m} \right) \cdot c \][/tex]
Where:
- [tex]\(L\)[/tex] is the lower boundary of the median class: [tex]\(30\)[/tex]
- [tex]\(N\)[/tex] is the total frequency: [tex]\(34\)[/tex]
- [tex]\(F\)[/tex] is the cumulative frequency before the median class: [tex]\(13\)[/tex]
- [tex]\(f_m\)[/tex] is the frequency of the median class: [tex]\(12\)[/tex]
- [tex]\(c\)[/tex] is the class interval width: [tex]\(10\)[/tex]
Substitute the values:
[tex]\[ \text{Median} = 30 + \left( \frac{17 - 13}{12} \right) \cdot 10 = 30 + \left( \frac{4}{12} \right) \cdot 10 = 30 + 3.33 \approx 33.33 \][/tex]
### Step 4: Calculate the Mode
To find the mode, identify the modal class, which is the class interval with the highest frequency. The frequency values are [tex]\(5, 8, 12, 6, 3\)[/tex]:
- The highest frequency is [tex]\(12\)[/tex] for the class interval [tex]\(30-40\)[/tex].
Calculate the mode using the formula:
[tex]\[ \text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \cdot c \][/tex]
Where:
- [tex]\(L\)[/tex] is the lower boundary of the modal class: [tex]\(30\)[/tex]
- [tex]\(f_1\)[/tex] is the frequency of the modal class: [tex]\(12\)[/tex]
- [tex]\(f_0\)[/tex] is the frequency of the class before the modal class: [tex]\(8\)[/tex]
- [tex]\(f_2\)[/tex] is the frequency of the class after the modal class: [tex]\(6\)[/tex]
- [tex]\(c\)[/tex] is the class interval width: [tex]\(10\)[/tex]
Substitute the values:
[tex]\[ \text{Mode} = 30 + \left( \frac{12 - 8}{2 \cdot 12 - 8 - 6} \right) \cdot 10 = 30 + \left( \frac{4}{24 - 14} \right) \cdot 10 = 30 + \left( \frac{4}{10} \right) \cdot 10 = 30 + 4 = 34.00 \][/tex]
### Summary
The ordered quantities (Mean, Median, Mode) for the given frequency distribution table are:
- Mean: 33.24
- Median: 33.33
- Mode: 34.00
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
