VIPSolutionsmake table nad solve mentioning all required calculation as in chegg calculate variance and standard deviation for this sample From the table (total ? = 28 n=28): Class (ft) Frequency 52.5–185.5 8 185.5–318.5 11 318.5–451.5 2 451.5–584.5 1 584.5–717.5 4 717.5–850.5 2
Question:
make table nad solve mentioning all required calculation as in chegg calculate variance and standard deviation for this sample From the table (total ? = 28 n=28): Class (ft) Frequency 52.5–185.5 8 185.5–318.5 11 318.5–451.5 2 451.5–584.5 1 584.5–717.5 4 717.5–850.5 2
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Asked by: nayan
Created at: 2026-02-04 08:36:52
Asked by: nayan
Created at: 2026-02-04 08:36:52
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# Variance and Standard Deviation Calculation for Grouped Data
Given sample (n = 28):
| Class Interval (ft) | Frequency (f) |
|---------------------|:------------:|
| 52.5–185.5 | 8 |
| 185.5–318.5 | 11 |
| 318.5–451.5 | 2 |
| 451.5–584.5 | 1 |
| 584.5–717.5 | 4 |
| 717.5–850.5 | 2 |
---
## Step 1: Find the Midpoints (x) of Each Class
\[
\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
\]
| Class Interval (ft) | Midpoint (x) | Frequency (f) |
|---------------------|:------------:|:-------------:|
| 52.5–185.5 | 119. | 8 |
| 185.5–318.5 | 252. | 11 |
| 318.5–451.5 | 385. | 2 |
| 451.5–584.5 | 518. | 1 |
| 584.5–717.5 | 651. | 4 |
| 717.5–850.5 | 784. | 2 |
---
## Step 2: Calculate \( f \times x \) and \( f \times x^2 \)
| Class Interval (ft) | Midpoint (x) | Frequency (f) | \(f \times x\) | \(f \times x^2\) |
|---------------------|:------------:|:-------------:|:--------------:|:-------------------:|
| 52.5–185.5 | 119. | 8 | 952. | 113,288. |
| 185.5–318.5 | 252. | 11 | 2772. | 697,752. |
| 318.5–451.5 | 385. | 2 | 770. | 296,450. |
| 451.5–584.5 | 518. | 1 | 518. | 268,324. |
| 584.5–717.5 | 651. | 4 | 2604. | 1,697,604. |
| 717.5–850.5 | 784. | 2 | 1568. | 1,229,056. |
---
## Step 3: Calculate Totals
\[
\sum f = 28
\]
\[
\sum (f \times x) = 952 + 2772 + 770 + 518 + 2604 + 1568 = 9,184
\]
\[
\sum (f \times x^2) = 113,288 + 697,752 + 296,450 + 268,324 + 1,697,604 + 1,229,056 = 4,302,474
\]
---
## Step 4: Calculate Mean (\(\bar{x}\))
\[
\bar{x} = \frac{\sum (f \times x)}{\sum f} = \frac{9184}{28} \approx 328.00
\]
---
## Step 5: Calculate Variance (\(s^2\)) and Standard Deviation (\(s\))
### Variance Formula (Sample):
\[
s^2 = \frac{\sum (f \times x^2) - \frac{[\sum (f \times x)]^2}{n}}{n-1}
\]
Plug values in:
\[
s^2 = \frac{4,302,474 - \frac{(9,184)^2}{28}}{27}
\]
\[
(9,184)^2 = 84,348,736
\]
\[
\frac{84,348,736}{28} = 3,012,454.857
\]
\[
s^2 = \frac{4,302,474 - 3,012,454.857}{27}
\]
\[
s^2 = \frac{1,290,019.143}{27}
\]
\[
s^2 \approx 47,779.23
\]
---
### Standard Deviation
\[
s = \sqrt{47,779.23} \approx 218.6
\]
---
## Final Answers
- **Sample Variance (\(s^2\))**: **47,779.23**
- **Sample Standard Deviation (\(s\))**: **218.6**
---
### Summary Table
| Statistic | Value |
|------------------|:----------:|
| Mean (\(\bar{x}\)) | 328.00 |
| Variance (\(s^2\)) | 47,779.23 |
| Std. Dev. (\(s\)) | 218.6 |
Question:
solution is incorrect try agauin
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Asked by: nayan
Created at: 2026-02-04 08:38:26
Asked by: nayan
Created at: 2026-02-04 08:38:26
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# Variance and Standard Deviation Calculation for Grouped Data
Given sample size \( n = 28 \):
| Class Interval (ft) | Frequency (\( f \)) |
|---------------------|:-------------------:|
| 52.5–185.5 | 8 |
| 185.5–318.5 | 11 |
| 318.5–451.5 | 2 |
| 451.5–584.5 | 1 |
| 584.5–717.5 | 4 |
| 717.5–850.5 | 2 |
---
## Step 1: Find the Midpoints (\( x \)) of Each Class
\[
\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
\]
| Class Interval (ft) | Midpoint (\( x \)) | Frequency (\( f \)) |
|---------------------|:-------------------:|:-------------------:|
| 52.5–185.5 | 119.0 | 8 |
| 185.5–318.5 | 252.0 | 11 |
| 318.5–451.5 | 385.0 | 2 |
| 451.5–584.5 | 518.0 | 1 |
| 584.5–717.5 | 651.0 | 4 |
| 717.5–850.5 | 784.0 | 2 |
---
## Step 2: Calculate \( f \times x \) and \( f \times x^2 \)
| Class Interval (ft) | Midpoint (\( x \)) | Frequency (\( f \)) | \( f \times x \) | \( f \times x^2 \) |
|---------------------|:-------------------:|:-------------------:|:-----------------:|:---------------------:|
| 52.5–185.5 | 119.0 | 8 | 952 | 113,288 |
| 185.5–318.5 | 252.0 | 11 | 2772 | 697,752 |
| 318.5–451.5 | 385.0 | 2 | 770 | 296,450 |
| 451.5–584.5 | 518.0 | 1 | 518 | 268,324 |
| 584.5–717.5 | 651.0 | 4 | 2604 | 1,697,604 |
| 717.5–850.5 | 784.0 | 2 | 1568 | 1,229,056 |
---
## Step 3: Calculate Totals
\[
\sum f = 28
\]
\[
\sum (f \times x) = 952 + 2772 + 770 + 518 + 2604 + 1568 = 9,184
\]
\[
\sum (f \times x^2) = 113,288 + 697,752 + 296,450 + 268,324 + 1,697,604 + 1,229,056 = 4,302,474
\]
---
## Step 4: Calculate Mean (\( \bar{x} \))
\[
\bar{x} = \frac{\sum (f \times x)}{\sum f} = \frac{9184}{28} \approx 328.00
\]
---
## Step 5: Calculate Variance (\( s^2 \)) and Standard Deviation (\( s \))
### Variance Formula (Sample):
\[
s^2 = \frac{\sum (f \times x^2) - \frac{[\sum (f \times x)]^2}{n}}{n-1}
\]
Substituting the values:
\[
s^2 = \frac{4,302,474 - \frac{(9,184)^2}{28}}{27}
\]
Calculating \( (9,184)^2 \):
\[
(9,184)^2 = 84,348,736
\]
\[
\frac{84,348,736}{28} \approx 3,012,454.857
\]
Now substituting back:
\[
s^2 = \frac{4,302,474 - 3,012,454.857}{27}
\]
\[
s^2 = \frac{1,290,019.143}{27}
\]
\[
s^2 \approx 47,779.23
\]
---
### Standard Deviation
\[
s = \sqrt{47,779.23} \approx 218.6
\]
---
## Final Answers
- **Sample Variance (\( s^2 \))**: **47,779.23**
- **Sample Standard Deviation (\( s \))**: **218.6**
---
### Summary Table
| Statistic | Value |
|-------------------|:----------:|
| Mean (\( \bar{x} \)) | 328.00 |
| Variance (\( s^2 \)) | 47,779.23 |
| Std. Dev. (\( s \)) | 218.6 |
Question:
corrct table is : Class ? f Midpoint ? x ? ? fx ? 2 x 2 ? ? 2 fx 2 52.5–185.5 8 119 952 14161 113288 185.5–318.5 11 252 2772 63504 698544 318.5–451.5 2 385 770 148225 296450 451.5–584.5 1 518 518 268324 268324 584.5–717.5 4 651 2604 423801 1,695,204 717.5–850.5 2 784 1568 614656 1,229,312 ∑ ? = 28 ∑f=28 ∑ ? ? = 9184 ∑fx=9184 ∑ ? ? 2 = 4 , 301 , 122 ∑fx 2 =4,301,122
This is AI generated Response.
Asked by: nayan
Created at: 2026-02-04 08:39:46
Asked by: nayan
Created at: 2026-02-04 08:39:46
Final Answer:
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# Variance and Standard Deviation Calculation for Grouped Data
Given sample size \( n = 28 \):
| Class Interval (ft) | Frequency (\( f \)) | Midpoint (\( x \)) | \( f \times x \) | \( x^2 \) | \( f \times x^2 \) |
|---------------------|:-------------------:|:-------------------:|:-----------------:|:---------:|:-------------------:|
| 52.5–185.5 | 8 | 119 | 952 | 14,161 | 113,288 |
| 185.5–318.5 | 11 | 252 | 2772 | 63,504 | 698,544 |
| 318.5–451.5 | 2 | 385 | 770 | 148,225 | 296,450 |
| 451.5–584.5 | 1 | 518 | 518 | 268,324 | 268,324 |
| 584.5–717.5 | 4 | 651 | 2604 | 423,801 | 1,695,204 |
| 717.5–850.5 | 2 | 784 | 1568 | 614,656 | 1,229,312 |
---
## Step 1: Calculate Totals
\[
\sum f = 28
\]
\[
\sum (f \times x) = 952 + 2772 + 770 + 518 + 2604 + 1568 = 9,184
\]
\[
\sum (f \times x^2) = 113,288 + 698,544 + 296,450 + 268,324 + 1,695,204 + 1,229,312 = 4,301,122
\]
---
## Step 2: Calculate Mean (\( \bar{x} \))
\[
\bar{x} = \frac{\sum (f \times x)}{\sum f} = \frac{9184}{28} \approx 328.00
\]
---
## Step 3: Calculate Variance (\( s^2 \)) and Standard Deviation (\( s \))
### Variance Formula (Sample):
\[
s^2 = \frac{\sum (f \times x^2) - \frac{[\sum (f \times x)]^2}{n}}{n-1}
\]
Substituting the values:
\[
s^2 = \frac{4,301,122 - \frac{(9,184)^2}{28}}{27}
\]
Calculating \( (9,184)^2 \):
\[
(9,184)^2 = 84,348,736
\]
\[
\frac{84,348,736}{28} = 3,012,454.857
\]
Now substituting back:
\[
s^2 = \frac{4,301,122 - 3,012,454.857}{27}
\]
\[
s^2 = \frac{1,288,667.143}{27}
\]
\[
s^2 \approx 47,713.24
\]
---
### Standard Deviation
\[
s = \sqrt{47,713.24} \approx 218.5
\]
---
## Final Answers
- **Sample Variance (\( s^2 \))**: **47,713.24**
- **Sample Standard Deviation (\( s \))**: **218.5**
---
### Summary Table
| Statistic | Value |
|-------------------|:----------:|
| Mean (\( \bar{x} \)) | 328.00 |
| Variance (\( s^2 \)) | 47,713.24 |
| Std. Dev. (\( s \)) | 218.5 |
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