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Graded Discussion Board for STA301 will be opened on February, 03, 2015 and will be remained open till February 04, 2015. The topic of this GDB will be,

“Why range is not considered as a good measure of variability? Why standard deviation is preferred over the other measures?”

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We are here with you hands in hands to facilitate your learning and do not appreciate the idea of copying or replicating solutions.

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# How to Measure Variability in Statistics

Statisticians use summary measures to describe the amount of variability or spread in a set of data. The most common measures of variability are the range, the Interquartile range (IQR), variance, and standard deviation.

The Range

The range is the difference between the largest and smallest values in a set of values.

For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. For this set of numbers, the range would be 11 - 1 or 10.

## The Interquartile Range (IQR)

The Interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.

Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.

• Q1 is the "middle" value in the first half of the rank-ordered data set.
• Q2 is the median value in the set.
• Q3 is the "middle" value in the second half of the rank-ordered data set.

The interquartile range is equal to Q3 minus Q1.

For example, consider the following numbers: 1, 3, 4, 5, 5, 6, 7, 11. Q1 is the middle value in the first half of the data set. Since there are an even number of data points in the first half of the data set, the middle value is the average of the two middle values; that is, Q1 = (3 + 4)/2 or Q1 = 3.5. Q3 is the middle value in the second half of the data set. Again, since the second half of the data set has an even number of observations, the middle value is the average of the two middle values; that is, Q3 = (6 + 7)/2 or Q3 = 6.5. The interquartile range is Q3 minus Q1, so IQR = 6.5 - 3.5 = 3.

The Variance

In a populationvariance is the average squared deviation from the population mean, as defined by the following formula:

σ2 = Σ ( Xi - μ )2 / N

Where σ2 is the population variance, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population.

Observations from a simple random sample can be used to estimate the variance of a population. For this purpose, sample variance is defined by slightly different formula, and uses a slightly different notation:

s2 = Σ ( xi - x )2 / ( n - 1 )

Where s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample. Using this formula, the sample variance can be considered an unbiased estimate of the true population variance. Therefore, if you need to estimate an unknown population variance, based on data from a simple random sample, this is the formula to use.

## The Standard Deviation

The standard deviation is the square root of the variance. Thus, the standard deviation of a population is:

σ = sqrt [ σ2] = sqrt [ Σ ( Xi - μ )2 / N ]

Where σ is the population standard deviation, σ2 is the population variance, μ is the population mean, Xi is the ith element from the population, and N is the number of elements in the population.

Statisticians often use simple random samples to estimate the standard deviation of a population, based on sample data. Given a simple random sample, the best estimate of the standard deviation of a population is:

s = sqrt [s2 ] = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ]

Where s is the sample standard deviation, s2 is the sample variance, x is the sample mean, xi is the ith element from the sample, and n is the number of elements in the sample.

Thanx Alman Rashid. Now kindly tell y statician prefer standard deviation as a good measure of variability instead of range??????

Solution:

The range, inert-quartile range and standard deviation are all measures of variables that show the amount of variability within the dataset. The range is the most evident measure of dispersion and it is the difference between the minimum and maximum values in a dataset, whereas the standard deviation is a measure that reviews the amount by which every value within a dataset varies from the mean. In actual fact it point out how securely the values in the dataset are bunched around the mean value.
Firstly, we will discuss that why the range in not considered as a good measure? This is why because the range is the simplest measure of variability to calculate but can be misleading if the dataset contains extreme values and the inter-quartile range reduces the problem by considering the variability within the middle 50 percent of the dataset.
Secondly, in making the comparison, the standard deviation is the most accurate measure of variability since is takes into account a measure of how every value in the dataset varies from the mean. Moreover, all the other values stay between those two bounds, the range stays the same. The standard deviation on the other hand, depends on every value in the dataset. Standard deviation is the square root of the variance, so the standard deviation is an especially useful measure of variability when the distribution is normal or approximately normal because the proportion of the distribution within a given number of standard deviations from the mean to be calculated. That’s why standard deviation is preferred over the other measures and this is the main point of subject question.

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thank u tehmina 4 sharing this.

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