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Chapter 3: Problem 4

Identify the symbols used for each of the following: (a) sample standard deviation; (b) population standard deviation; (c) sample variance; (d) population variance.

Short Answer

Expert verified
(a) s, (b) \( \sigma \), (c) s虏, (d) \( \sigma^2 \)

Step by step solution

01

Identify the symbol for Sample Standard Deviation

The symbol for the sample standard deviation is `s`. This symbol is used to represent the standard deviation of a sample dataset.
02

Identify the symbol for Population Standard Deviation

The symbol for the population standard deviation is \( \sigma \). This symbol represents the standard deviation of an entire population.
03

Identify the symbol for Sample Variance

The symbol for the sample variance is \( s^2 \). Variance is the square of the standard deviation, and for a sample, we use the sample standard deviation squared.
04

Identify the symbol for Population Variance

The symbol for the population variance is \( \sigma^2 \). This is the variance of the entire population, calculated as the square of the population standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Standard Deviation
The sample standard deviation measures the amount of variation or dispersion in a sample set of data. Symbolized by `s`, it is used when you are analyzing a part of the entire population. Calculating the sample standard deviation helps to understand how spread out the sample data points are from the mean (average) of the sample.

To calculate it, you take the square root of the sample variance. The formula for the sample standard deviation is: \[ s = \sqrt{\dfrac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2} \] Here, \overline{x} is the sample mean, \( x_i \) represents each sample point, and \dfrac{1}{n-1} \ is the adjustment made to account for the degrees of freedom.
Population Standard Deviation
Population standard deviation, denoted by the Greek letter `\sigma`, measures the dispersion of data points in the entire population. It tells you how much the values in a population vary from the mean of the population. Unlike the sample standard deviation, it includes all members of a population.

The formula to calculate population standard deviation is: \[ \sigma = \sqrt{\dfrac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} \] Here, \mu is the population mean, \ x_i are the data points in the population, and N is the total number of data points in the population.
Sample Variance
Sample variance, symbolized by `s^2`, quantifies the degree of variation or dispersion of sample data points. It is the average of the squared differences from the sample mean. The higher the variance, the more spread out the data points are.

To calculate the sample variance, use the formula: \[ s^2 = \dfrac{1}{n-1} \sum_{i=1}^{n} (x_i - \overline{x})^2 \] Here, \overline{x} is the sample mean and `n` is the number of data points in the sample. By squaring the differences, you eliminate any negative values, which provides a clearer measure of dispersion.
Population Variance
Population variance, denoted as `\sigma^2`, provides an understanding of the spread of the entire set of data points in a population. It is the average of the squared differences from the population mean, reflecting the overall variability within the population.

The formula to calculate the population variance is: \[ \sigma^2 = \dfrac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2 \] where \mu is the population mean, \ x_i are the individual data points, and N is the total number of data points. The population variance helps researchers and statisticians understand the variability and consistency of the entire population dataset.

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Most popular questions from this chapter

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