Questions and Answers
What is the primary advantage of using variance over range in measuring dispersion?
Variance takes into account all the observations
Which measure of dispersion is most sensitive to the presence of outliers in a dataset?
Range
What is the formula to calculate the range of a dataset?
Range = Max - Min
Which of the following is a limitation of using range as a measure of dispersion?
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What is the primary difference between range and variance as measures of dispersion?
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Why is variance a more comprehensive measure of dispersion than range?
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What happens to the range of a dataset when an outlier is introduced?
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Which of the following is an advantage of using standard deviation over range?
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What is the unit of variance in a dataset of weights of students measured in kg?
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If the sample standard deviation is 14.52, what is the sample variance?
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What is the effect of adding a constant to a dataset on the standard deviation?
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What is the unit of standard deviation in a dataset of ages of students measured in years?
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If the population standard deviation of a dataset is σ, what is the population standard deviation of the dataset after adding a constant c to each observation?
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What is the formula for population standard deviation?
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What happens to the standard deviation of a dataset if each observation is increased by 10?
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What is the relationship between the standard deviation and the variance of a dataset?
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What is the relationship between the new population variance and the old population variance when each observation is multiplied by a constant c?
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What is the formula for the new population variance when each observation is multiplied by a constant c?
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What is the effect on the population variance when each observation is multiplied by a constant c?
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What is the relationship between the population variance and the standard deviation?
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What is the formula for the population variance?
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What is the effect on the population standard deviation when each observation is multiplied by a constant c?
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What is the relationship between the new population standard deviation and the old population standard deviation when each observation is multiplied by a constant c?
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What is the purpose of calculating the population variance?
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Study Notes
Variance and Standard Deviation
- The sample standard deviation is the square root of the sample variance.
- The units of variance are square units of the original variable, while the units of standard deviation are the same as the original data.
Adding a Constant
- If a constant is added to each observation, the new standard deviation is equal to the old standard deviation.
- Proof: σnew = σold
- This is true for both population and sample standard deviations.
Multiplying by a Constant
- If each observation is multiplied by a constant, the new standard deviation is equal to the old standard deviation multiplied by the constant.
- Proof: σnew = c × σold
- This is true for both population and sample standard deviations.
Standard Deviation
- Standard deviation is a measure of dispersion and is the square root of the variance.
- Other measures of dispersion include range, variance, interquartile range, and standard deviation.
Range
- The range of a dataset is the difference between its largest and smallest values.
- Formula: Range = Max - Min
- The range is sensitive to outliers as it only considers the minimum and maximum values of the dataset.
Effect of Outliers on Range
- The range can be significantly affected by a single outlier in the dataset.
- Example: A single outlier can change the range from 4 to 14 in a dataset.
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