Mean, Standard Deviation, and Standard Error Calculations Calculate the sample mean, sample standard deviation, standard error of the mean (SEM), and 95% confidence interval (2 x SEM) rounded to three decimal places and enter the numbers below You will find instructions and formulas for calculating these statistics under the “Calculations” tab If you already performed these calculations in Module 3, press “Auto Calculate” and the table will be automatically filled for you (Note that using the auto calculate feature will be recorded in your progress report) A. cristatellus A. cooki Individual Dewlap Color Mean Calculations Individual Dewlap Color Mean Sample Mean Calculations Sample Mean Download Data 2.5 Auto Calculate 2 5.5 Sample SD 2 Sample SD Clear Calculations 5 3 4.5 SEM SEM 4 5.5 4 1.5 95% CI 95% CI Save & Continue 5 5.5 3.5
The Correct Answer and Explanation is :
To calculate the sample mean, sample standard deviation (SD), standard error of the mean (SEM), and the 95% confidence interval (CI), we use the following formulas:
- Sample Mean ((\bar{x})):
[
\bar{x} = \frac{\sum x_i}{n}
]
where (x_i) are the individual values, and (n) is the number of samples. - Sample Standard Deviation (SD):
[
SD = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}
] - Standard Error of the Mean (SEM):
[
SEM = \frac{SD}{\sqrt{n}}
] - 95% Confidence Interval (CI):
[
95\% CI = 2 \times SEM
]
Given Data:
- Values: 2, 5.5, 5, 3, 4.5, 4, 5.5, 1.5, 5, 5.5, 3.5
Step-by-Step Calculation:
- Sample Mean ((\bar{x})):
[
\bar{x} = \frac{2 + 5.5 + 5 + 3 + 4.5 + 4 + 5.5 + 1.5 + 5 + 5.5 + 3.5}{11} = \frac{49}{11} = 4.455
] - Sample Standard Deviation (SD):
First, calculate deviations from the mean and square them:
[
(x_i – \bar{x})^2 = (2-4.455)^2, (5.5-4.455)^2, \dots
]
Sum these squared deviations, divide by (n-1 = 10), and take the square root.
After calculation, (SD = 1.405). - Standard Error of the Mean (SEM):
[
SEM = \frac{SD}{\sqrt{n}} = \frac{1.405}{\sqrt{11}} = 0.424
] - 95% Confidence Interval (CI):
[
95\% CI = 2 \times SEM = 2 \times 0.424 = 0.848
]
Final Results:
- Sample Mean: (4.455)
- Sample SD: (1.405)
- SEM: (0.424)
- 95% CI: (0.848)
Explanation:
The sample mean ((\bar{x})) represents the average value of the dataset. The standard deviation measures how much the individual data points vary from the mean. SEM estimates the variability of the mean if we were to repeatedly sample. A 95% CI provides a range likely containing the true population mean, assuming normality.