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Questions and Answers
What does the standard deviation measure?
What does the standard deviation measure?
The standard deviation is _____ when the data are all concentrated close to the mean, exhibiting little variation or spread.
The standard deviation is _____ when the data are all concentrated close to the mean, exhibiting little variation or spread.
small
The standard deviation is ______ when the data values are more spread out from the mean, exhibiting more variation.
The standard deviation is ______ when the data values are more spread out from the mean, exhibiting more variation.
larger
What is the formula for calculating the value based on the standard deviation?
What is the formula for calculating the value based on the standard deviation?
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How is the sample standard deviation represented?
How is the sample standard deviation represented?
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What is a deviation in statistics?
What is a deviation in statistics?
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The sample standard deviation is a good estimate of the population standard deviation if the sample has the same characteristics as the population.
The sample standard deviation is a good estimate of the population standard deviation if the sample has the same characteristics as the population.
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What is variance?
What is variance?
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How are the population variance and population standard deviation related?
How are the population variance and population standard deviation related?
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How much the statistic varies from one sample to another is known as the ____ ____ of a statistic.
How much the statistic varies from one sample to another is known as the ____ ____ of a statistic.
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What does the standard error of the mean measure?
What does the standard error of the mean measure?
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What is the notation for the standard error of the mean?
What is the notation for the standard error of the mean?
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If you add the deviations, the sum is always ____.
If you add the deviations, the sum is always ____.
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By squaring the deviations, the sum will also be ____.
By squaring the deviations, the sum will also be ____.
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#ofSTDEVs is often called a '-____'; we can use the symbol z.
#ofSTDEVs is often called a '-____'; we can use the symbol z.
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Chebyshev's Rule states that at least 95% of the data is within four and a half standard deviations of the mean for any data set.
Chebyshev's Rule states that at least 95% of the data is within four and a half standard deviations of the mean for any data set.
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The Empirical Rule applies only when the data's distribution is bell-shaped and symmetric.
The Empirical Rule applies only when the data's distribution is bell-shaped and symmetric.
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Study Notes
Standard Deviation
- Measures how far data values deviate from the mean, indicating variation in a data set.
- A small standard deviation indicates data points are concentrated close to the mean.
- A large standard deviation signifies data points are more spread out from the mean.
Formulas and Definitions
- General formula: value = mean + (# of STDEV) × (standard deviation).
- For samples: x = x̄ + (# of STDEV) × s; for populations: x = μ + (# of STDEV) × σ.
- Deviation refers to the difference between a data point and the mean, represented as x - mean.
Variance
- Defined as the average of the squares of the deviations from the mean.
- Population variance symbolized by σ²; sample variance symbolized by s².
- Standard deviation is the square root of variance.
Sampling Variability
- Variation in a statistic from sample to sample is known as sampling variability.
- The standard error of the mean quantifies this variability and illustrates the precision of sample means.
Standard Error
- Formula for standard error of the mean: σ/√n, where σ is the population standard deviation and n is the sample size.
Properties of Deviations
- The sum of deviations from the mean is always zero.
- Squaring deviations results in positive numbers, leading to a positive sum.
Z-Score
- The term "# of STDEVs" is also referred to as the z-score, represented by the symbol z.
Chebyshev's Rule
- At least 75% of data lies within two standard deviations of the mean.
- At least 89% of data lies within three standard deviations.
- At least 95% of data lies within 4.5 standard deviations.
Empirical Rule
- For bell-shaped and symmetric distributions:
- About 68% of data falls within one standard deviation of the mean.
- About 95% of data falls within two standard deviations.
- More than 99% of data falls within three standard deviations.
- This rule is specific to normal (Gaussian) distributions.
Studying That Suits You
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Description
Test your knowledge of key terms in statistics with these flashcards focused on Chapter 2.7. Learn about standard deviation and how it quantifies data variation. This quiz is perfect for students looking to reinforce their understanding of statistical concepts.
