Questions and Answers
What is the measure of dispersion that requires the least amount of steps to calculate?
Range
Explain how to find the range of a data set.
The range of a data set is the difference between the maximum and minimum data entries in the set.
What is an advantage of using the range as a measure of variation?
What is a disadvantage of using the range as a measure of variation?
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Find the range of the data set from a leaf stem histogram where the lowest number is 23 and the highest number is 97.
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What does the symbol sigma represent?
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What is the range of the April precipitation amounts if the highest is 4 inches and the lowest is 0.8 inches?
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Explain the relationship between variance and standard deviation.
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Can either variance or standard deviation be negative?
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What can be said about a set of data with a standard deviation of zero?
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Can a standard deviation be negative?
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To which distributions does the empirical rule apply?
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In a normal distribution, approximately what percentage of the area under the normal curve is within two standard deviations of the mean?
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According to the empirical rule, what percent of the area under the normal curve is between the mean and one standard deviation above the mean?
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What percentage of the area under the normal curve is more than one standard deviation from the mean?
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Use the empirical rule to determine what percentage of people has an IQ score between 52 and 148, if the mean is 100 and the standard deviation is 16.
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What percentage of people has an IQ score less than 84 or greater than 116?
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What percentage of people has an IQ score greater than 116?
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Study Notes
Range
- The range is the measure of dispersion that requires the least steps to calculate.
- To find the range, subtract the minimum value from the maximum value in a dataset.
Advantages and Disadvantages of Range
- Advantage: Simple to compute.
- Disadvantage: Only considers two entries (maximum and minimum) and may not accurately reflect the data's overall variation.
Example Calculation of Range
- For a dataset ranging from 23 to 97, the calculated range is 74 (97 - 23).
Symbols and Definitions
- Sigma (σ) represents the population standard deviation.
Calculating Range Example
- Given highest precipitation of 4 inches and lowest of 0.8 inches, the range is 3.2 inches.
Variance and Standard Deviation
- Standard deviation is the positive square root of variance.
- Both variance and standard deviation cannot be negative since squared deviations are always non-negative.
Population vs. Sample
- Determine whether to use σ (population) or S (sample) based on the context of the dataset.
Zero Standard Deviation
- A standard deviation of zero indicates all observations in the dataset are the same value, resulting in no variation.
Standard Deviation Properties
- Standard deviation cannot be negative as it measures data spread from the mean.
Empirical Rule
- The empirical rule applies to normal distributions and estimates percentages within specific standard deviations.
- Approximately 95% of data falls within two standard deviations from the mean.
Normal Distribution and Standard Deviations
- About 68% of data lies within one standard deviation from the mean.
- 34% of the area under the curve is between the mean and one standard deviation above the mean.
Area Beyond One Standard Deviation
- 32% of data lies more than one standard deviation away from the mean, calculated as 100% - 68%.
IQ Test Scores Analysis
- IQ scores have a mean of 100 and a standard deviation of 16.
- Approximately 99.7% of individuals have IQ scores between 52 and 148, which encompasses three standard deviations.
- 32% of individuals score less than 84 or greater than 116 (outside one standard deviation).
- 16% of individuals have scores greater than 116 (half of the 32% beyond one standard deviation).
Additional Note
- No content provided for Card 16 to summarize.
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Description
This quiz focuses on key concepts related to the range in statistics. It includes definitions and an explanation of how to calculate the range of a data set. Ideal for students working on their statistics homework.
