Computational Formula for Standard Deviation

Oct 13, 2025 by ludoplayers7.com 135 views

Understanding the Problem

The given data represents the weights (in grams) of a random sample of 48 M&M plain candies. We are required to complete parts (a) through (f) using the computational formula for standard deviation.

Computational Formula for Standard Deviation

The computational formula for standard deviation is given by:

${ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} }$

where:

${ s }$ is the sample standard deviation
${ x_i }$ is the individual data point
${ \bar{x} }$ is the sample mean
${ n }$ is the number of data points

Part (a): Calculate the Sample Mean

To calculate the sample mean, we need to add up all the weights and divide by the number of data points.

Weight (in grams)
2.5
2.8
3.1
3.4
3.7
4.0
4.3
4.6
4.9
5.2
5.5
5.8
6.1
6.4
6.7
7.0
7.3
7.6
7.9
8.2
8.5
8.8
9.1
9.4
9.7
10.0
10.3
10.6
10.9
11.2
11.5
11.8
12.1
12.4
12.7
13.0
13.3
13.6
13.9
14.2
14.5
14.8
15.1
15.4
15.7
16.0
16.3
16.6
16.9
17.2
17.5
17.8
18.1
18.4
18.7
19.0
19.3
19.6
19.9

Calculating the Sample Mean

To calculate the sample mean, we add up all the weights and divide by the number of data points.

import numpy as np
weights = np.array([2.5, 2.8, 3.1, 3.4, 3.7, 4.0, 4.3, 4.6, 4.9, 5.2, 5.5, 5.8, 6.1, 6.4, 6.7, 7.0, 7.3, 7.6, 7.9, 8.2, 8.5, 8.8, 9.1, 9.4, 9.7, 10.0, 10.3, 10.6, 10.9, 11.2, 11.5, 11.8, 12.1, 12.4, 12.7, 13.0, 13.3, 13.6, 13.9, 14.2, 14.5, 14.8, 15.1, 15.4, 15.7, 16.0, 16.3, 16.6, 16.9, 17.2, 17.5, 17.8, 18.1, 18.4, 18.7, 19.0, 19.3, 19.6, 19.9])
sample_mean = np.mean(weights)
print("Sample Mean:", sample_mean)

Part (b): Calculate the Deviations from the Sample Mean

To calculate the deviations from the sample mean, we subtract the sample mean from each data point.

# Calculate the deviations from the sample mean deviations = weights - sample_mean

print("Deviations from Sample Mean:", deviations)

Part (c): Calculate the Squared Deviations

To calculate the squared deviations, we square each deviation.

# Calculate the squared deviations squared_deviations = deviations ** 2

print("Squared Deviations:", squared_deviations)

Part (d): Calculate the Sum of Squared Deviations

To calculate the sum of squared deviations, we add up all the squared deviations.

# Calculate the sum of squared deviations
sum_squared_deviations = np.sum(squared_deviations)
print("Sum of Squared Deviations:", sum_squared_deviations)

Part (e): Calculate the Sample Variance

To calculate the sample variance, we divide the sum of squared deviations by the number of data points minus one.

# Calculate the sample variance
sample_variance = sum_squared_deviations / (len(weights) - 1)
print("Sample Variance:", sample_variance)

Part (f): Calculate the Sample Standard Deviation

To calculate the sample standard deviation, we take the square root of the sample variance.

# Calculate the sample standard deviation
sample_standard_deviation = np.sqrt(sample_variance)
print("Sample Standard Deviation:", sample_standard_deviation)

Conclusion

Frequently Asked Questions

Q: What is the sample standard deviation?

A: The sample standard deviation is a measure of the spread or dispersion of the data from the sample mean. It is calculated using the computational formula:

${ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} }$

where:

${ s }$ is the sample standard deviation
${ x_i }$ is the individual data point
${ \bar{x} }$ is the sample mean
${ n }$ is the number of data points

Q: How do I calculate the sample standard deviation?

A: To calculate the sample standard deviation, you need to follow these steps:

Calculate the sample mean
Calculate the deviations from the sample mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: What is the difference between the sample standard deviation and the population standard deviation?

A: The sample standard deviation is used when the data is a sample from a larger population, while the population standard deviation is used when the data is the entire population. The formula for the sample standard deviation is:

${ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} }$

while the formula for the population standard deviation is:

${ \sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} }$

where:

${ \sigma }$ is the population standard deviation
${ x_i }$ is the individual data point
${ \bar{x} }$ is the population mean
${ N }$ is the number of data points in the population

Q: How do I interpret the sample standard deviation?

A: The sample standard deviation is a measure of the spread or dispersion of the data from the sample mean. A small sample standard deviation indicates that the data points are close to the sample mean, while a large sample standard deviation indicates that the data points are spread out.

Q: What is the relationship between the sample standard deviation and the sample mean?

A: The sample standard deviation is a measure of the spread or dispersion of the data from the sample mean. The sample standard deviation is always positive, while the sample mean can be positive or negative.

Q: Can I use the sample standard deviation to make predictions about the population?

A: The sample standard deviation can be used to make predictions about the population, but it is not a guarantee. The sample standard deviation is a measure of the spread or dispersion of the data from the sample mean, and it can be affected by outliers or other factors that may not be representative of the population.

Q: How do I calculate the sample standard deviation using a calculator or computer software?

A: To calculate the sample standard deviation using a calculator or computer software, you can use the following steps:

Enter the data into the calculator or computer software
Select the "statistics" or "descriptive statistics" option
Choose the "sample standard deviation" option
The calculator or computer software will calculate the sample standard deviation for you

Q: What are some common mistakes to avoid when calculating the sample standard deviation?

A: Some common mistakes to avoid when calculating the sample standard deviation include:

Not using the correct formula
Not using the correct data
Not checking for outliers or other factors that may affect the sample standard deviation
Not using the correct calculator or computer software

Q: Can I use the sample standard deviation to compare two or more groups?

A: Yes, you can use the sample standard deviation to compare two or more groups. The sample standard deviation can be used to determine if there are any significant differences between the groups.

Q: How do I calculate the sample standard deviation for a grouped data set?

A: To calculate the sample standard deviation for a grouped data set, you can use the following steps:

Calculate the midpoint of each group
Calculate the frequency of each group
Calculate the squared deviations for each group
Calculate the sum of squared deviations for each group
Calculate the sample variance for each group
Calculate the sample standard deviation for each group

Q: Can I use the sample standard deviation to make inferences about the population?

A: Yes, you can use the sample standard deviation to make inferences about the population. The sample standard deviation can be used to determine if there are any significant differences between the sample and the population.

Q: How do I calculate the sample standard deviation for a time series data set?

A: To calculate the sample standard deviation for a time series data set, you can use the following steps:

Calculate the mean of the time series data
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to predict future values?

A: Yes, you can use the sample standard deviation to predict future values. The sample standard deviation can be used to determine the range of possible values for future observations.

Q: How do I calculate the sample standard deviation for a non-normal distribution?

A: To calculate the sample standard deviation for a non-normal distribution, you can use the following steps:

Calculate the median of the data
Calculate the interquartile range (IQR)
Calculate the squared deviations from the median
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to compare two or more non-normal distributions?

A: Yes, you can use the sample standard deviation to compare two or more non-normal distributions. The sample standard deviation can be used to determine if there are any significant differences between the distributions.

Q: How do I calculate the sample standard deviation for a censored data set?

A: To calculate the sample standard deviation for a censored data set, you can use the following steps:

Calculate the mean of the censored data
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to make inferences about the population for a censored data set?

A: Yes, you can use the sample standard deviation to make inferences about the population for a censored data set. The sample standard deviation can be used to determine if there are any significant differences between the sample and the population.

Q: How do I calculate the sample standard deviation for a data set with missing values?

A: To calculate the sample standard deviation for a data set with missing values, you can use the following steps:

Calculate the mean of the data set
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to make inferences about the population for a data set with missing values?

A: Yes, you can use the sample standard deviation to make inferences about the population for a data set with missing values. The sample standard deviation can be used to determine if there are any significant differences between the sample and the population.

Q: How do I calculate the sample standard deviation for a data set with outliers?

A: To calculate the sample standard deviation for a data set with outliers, you can use the following steps:

Calculate the mean of the data set
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to make inferences about the population for a data set with outliers?

A: Yes, you can use the sample standard deviation to make inferences about the population for a data set with outliers. The sample standard deviation can be used to determine if there are any significant differences between the sample and the population.

Q: How do I calculate the sample standard deviation for a data set with a non-linear relationship?

A: To calculate the sample standard deviation for a data set with a non-linear relationship, you can use the following steps:

Calculate the mean of the data set
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation

Q: Can I use the sample standard deviation to make inferences about the population for a data set with a non-linear relationship?

A: Yes, you can use the sample standard deviation to make inferences about the population for a data set with a non-linear relationship. The sample standard deviation can be used to determine if there are any significant differences between the sample and the population.

Q: How do I calculate the sample standard deviation for a data set with a non-constant variance?

A: To calculate the sample standard deviation for a data set with a non-constant variance, you can use the following steps:

Calculate the mean of the data set
Calculate the deviations from the mean
Calculate the squared deviations
Calculate the sum of squared deviations
Calculate the sample variance
Calculate the sample standard deviation