MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Calculating Variance and Standard Error for Bird Weight Data
#### Bird Weight Data
The data provided for the bird weights is obtained from a publication by Baldwin & Kendeigh written in 1938. The weights are given in grams:
- 11.4 grams
- 11.9 grams
- 12.4 grams
- 12.9 grams
- 14.1 grams
#### Variance and Standard Error Calculation
To calculate the variance (\(s^2\)) for the given data, we can use the following formula:
\[
s^2 = \frac{n}{n - 1} \times (\mathbb{E}[x^2] - [\mathbb{E}(x)]^2)
\]
where:
- \( n \) is the number of data points,
- \(\mathbb{E}[x^2]\) is the expected value of the square of the data points,
- \(\mathbb{E}(x)\) is the expected value of the data points.
The above formula helps in estimating the variance by using the sample data in such a way that it corrects for biases.
##### Detailed Explanation:
1. **Expected Value of \(x\)** (\(\mathbb{E}(x)\)): It's the mean of the sample data.
2. **Expected Value of \(x^2\)** (\(\mathbb{E}[x^2]\)): This is calculated by taking the mean of each data point squared.
Next, we adjust our variance by multiplying by \(\frac{n}{n-1}\) to deal with bias in the sample variance estimate.
By calculating the variance first, we can then infer the standard error if needed by taking the square root of the variance divided by the number of samples.
No graphical representation is provided in the original document; the formulation is purely numerical and mathematical.](https://content.bartleby.com/qna-images/question/eceb8058-c7a9-4ec1-9952-bafa6949c626/bdb391c4-1180-4cee-8468-fa1c2719adf1/o008gfz_processed.jpeg)
Transcribed Image Text:### Calculating Variance and Standard Error for Bird Weight Data
#### Bird Weight Data
The data provided for the bird weights is obtained from a publication by Baldwin & Kendeigh written in 1938. The weights are given in grams:
- 11.4 grams
- 11.9 grams
- 12.4 grams
- 12.9 grams
- 14.1 grams
#### Variance and Standard Error Calculation
To calculate the variance (\(s^2\)) for the given data, we can use the following formula:
\[
s^2 = \frac{n}{n - 1} \times (\mathbb{E}[x^2] - [\mathbb{E}(x)]^2)
\]
where:
- \( n \) is the number of data points,
- \(\mathbb{E}[x^2]\) is the expected value of the square of the data points,
- \(\mathbb{E}(x)\) is the expected value of the data points.
The above formula helps in estimating the variance by using the sample data in such a way that it corrects for biases.
##### Detailed Explanation:
1. **Expected Value of \(x\)** (\(\mathbb{E}(x)\)): It's the mean of the sample data.
2. **Expected Value of \(x^2\)** (\(\mathbb{E}[x^2]\)): This is calculated by taking the mean of each data point squared.
Next, we adjust our variance by multiplying by \(\frac{n}{n-1}\) to deal with bias in the sample variance estimate.
By calculating the variance first, we can then infer the standard error if needed by taking the square root of the variance divided by the number of samples.
No graphical representation is provided in the original document; the formulation is purely numerical and mathematical.
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