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Chapter Analysis
Intermediate29 pages • EnglishQuick Summary
This chapter focuses on the topic of dispersion in statistics, covering various measures such as range, mean deviation, variance and standard deviation. These measures assess the spread of data around a central value. The chapter explains both ungrouped and grouped data and provides various methods like the direct method and the step-deviation method to calculate these measures. The importance of standard deviation as a measure of dispersion is also highlighted.
Key Topics
- •Measures of Dispersion
- •Variance and Standard Deviation
- •Mean Deviation
- •Range
- •Calculating variance for grouped data
- •Difference between deviation measures
- •Standard deviation formulas
Learning Objectives
- ✓Understand the concept of dispersion in data.
- ✓Calculate mean deviation, variance, and standard deviation.
- ✓Distinguish between different measures of dispersion.
- ✓Apply statistical methods to both ungrouped and grouped data.
- ✓Use shortcut methods to simplify statistical calculations.
Questions in Chapter
The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
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The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14. Find the remaining two observations.
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The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
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Given that x is the mean and σ2 is the variance of n observations x1, x2, ...,xn. Prove that the mean and variance of the observations ax1, ax2, ax3, ...., axn are ax and a2σ2, respectively, (a ≠ 0).
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The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases: (i) If wrong item is omitted. (ii) If it is replaced by 12.
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The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
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Additional Practice Questions
Explain the difference between mean deviation and standard deviation.
mediumAnswer: Mean deviation calculates the dispersion of data points by taking the average of absolute deviations from a central point, while standard deviation uses the square root of variance and gives more weight to larger deviations due to squaring differences from the mean.
Why is variance always non-negative for any set of observations?
easyAnswer: Variance is always non-negative because it is the average of the squared differences from the mean, and squaring any real number always results in a non-negative value.
How would you calculate the variance of a set of data points using the shortcut method?
mediumAnswer: In the shortcut method for variance, you can use step-deviation method which simplifies the calculations by assuming an assumed mean and calculating variances using reduced scales of deviations.
Provide an example where the mean deviation is more informative than the standard deviation.
hardAnswer: In datasets with extreme outliers, mean deviation can sometimes provide clearer insight into spread around a median, as it does not square the deviations, potentially offering a better picture if the data is not normally distributed.
If a dataset has zero variance, what can we conclude about the data points?
easyAnswer: If the variance is zero, it indicates that all data points are identical because there is no spread or dispersion from the mean.