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Chapter 3: Problem 4

What symbol is used for the standard deviation when it is a sample statistic? What symbol is used for the standard deviation when it is a population parameter?

Short Answer

Expert verified
For sample standard deviation, use "\( s \)"; for population standard deviation, use "\( \sigma \)".

Step by step solution

01

Understanding Standard Deviation Symbols

In statistics, different symbols are used to represent the standard deviation depending on whether we are dealing with a sample or a population. The type of data set determines the symbol.
02

Identifying Sample Standard Deviation

The symbol used to denote the standard deviation of a sample is "\( s \)". It is used when you calculate standard deviation from a subset of the entire population.
03

Identifying Population Standard Deviation

The symbol used for standard deviation when it applies to an entire population is "\( \sigma \)". This is the Greek letter sigma, and it represents the standard deviation for the entire set of data in the population.

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Key Concepts

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

Sample Statistic
A sample statistic is a value calculated from a subset of a larger population. It helps us make inferences about the entire population without needing to measure every individual in that group.
  • In practice, collecting data from an entire population is often too costly and time-consuming.
  • By using samples, we can estimate various population metrics, such as the mean, variance, and standard deviation.
  • The standard deviation in a sample is denoted by the symbol \( s \).
This symbol indicates that it is not based on the total data set but just a part of it. The sample standard deviation provides us with an idea of how spread out values are within the sample.In essence, statistics like the sample standard deviation are tools that enable us to make educated guesses about the broader group based on just a part of it.
Population Parameter
A population parameter, on the other hand, refers to any characteristic of a population that is usually unknown and needs to be estimated. It typically involves the entire group that a researcher is observing or interested in understanding.
  • The parameter provides a complete insight into the characteristic being measured, offering concrete values that describe the whole group.
  • The standard deviation of a population is indicated by the Greek letter \( \sigma \), representing the actual spread of data points around the mean in the entire population.
  • The population parameter depicts the exact value, unlike the sample statistic, which is just an estimate.
Knowing the difference between a sample statistic and a population parameter is crucial for conducting an accurate analysis and drawing reliable conclusions from data.
Greek Letter Sigma
The Greek letter sigma, \( \sigma \), plays a significant role in statistics, especially when talking about standard deviation. It's essential to recognize what this symbol represents and when to use it.
  • \( \sigma \) is always used when referring to the standard deviation of an entire population.
  • The choice of using a Greek letter signifies its status as a parameter, something we rarely know with complete accuracy unless the entire population is assessed.
Sigma's usage denotes the measurement's validity for the whole data set under consideration, indicating no uncertainty like when dealing with samples. In summary, understanding how and when to use \( \sigma \) is fundamental to conducting statistical analyses, as it distinguishes between data characteristics measured from all existing data points in the comparison to those estimated from just some.

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Most popular questions from this chapter

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 5,9,10,11,15 (a) Use the defining formula, the computation formula, or a calculator to compute \(s\) (b) Multiply each data value by 5 to obtain the new data set 25,45,50,55 75. Compute \(s\) (c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant \(c ?\) (d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be \(s=3.1\) miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations? Given 1 mile \(\approx 1.6\) kilometers, what is the standard deviation in kilometers?

What was the age distribution of prehistoric Native Americans? Extensive anthropologic studies in the southwestern United States gave the following information about a prehistoric extended family group of 80 members on what is now the Navajo Reservation in northwestern New Mexico (Source: Based on information taken from Prehistory in the Navajo Reservation District, by F. W. Eddy, Museum of New Mexico Press). $$\begin{array}{l|cccc} \hline \text { Age range (years) } & 1-10^{\star} & 11-20 & 21-30 & 31 \text { and over } \\ \hline \text { Number of individuals } & 34 & 18 & 17 & 11 \\ \hline \end{array}$$ For this community, estimate the mean age expressed in years, the sample variance, and the sample standard deviation. For the class 31 and over, use 35.5 as the class midpoint.

Do bonds reduce the overall risk of an investment portfolio? Let \(x\) be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let \(y\) be a random variable representing annual return for Vanguard Balanced Index \((60 \% \text { stock and } 40 \%\) bond). For the past several years, we have the following data (Reference: Morning star Research Group, Chicago). $$\begin{array}{ccccccccccc} x: & 11 & 0 & 36 & 21 & 31 & 23 & 24 & -11 & -11 & -21 \\ y: & 10 & -2 & 29 & 14 & 22 & 18 & 14 & -2 & -3 & -10 \end{array}$$ (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y,\) and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the two funds. (d) Compute the coefficient of variation for each fund. Use the coefficients of variation to compare the two funds. If \(s\) represents risks and \(\bar{x}\) represents expected return, then \(s / \bar{x}\) can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller CV better? Explain.

For a given data set in which not all data values are equal, which value is smaller, \(s\) or \(\sigma ?\) Explain.

Basic Computation: Weighted Average Find the weighted average of a data set where 10 has a weight of \(5 ; 20\) has a weight of \(3 ; 30\) has a weight of 2
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