Question

Answer the following questions about variability of data sets: How would you describe the variance and standard deviation in words, rather than a formula? Think of what you are calculating and how it might be useful in describing a variable. What is the primary advantage of using the inter-quartile range compared with the range when describing the variability of a variable? Can the standard deviation ever be larger than the variance? Explain. Can the variance ever be negative? Why or why not? Show the formula for the Coefficient of Variation and explain what it is and how it can be useful in comparing the variability of different variables.

Answer 1

How would you describe the variance and standard deviation in words, rather than a formula? Think of what you are calculating and how it might be useful in describing a variable.

The variance and standard deviations are the numerical coefficients for measuring the spread or variation in the given data or sample. Variance and standard deviation both are used for same purpose and the term standard deviation used as the square root of variance and it is used for reliable convenience for representing variance. It is very useful in describing the variable because it gives us the idea about spread or variation exists in the data.

What is the primary advantage of using the inter-quartile range compared with the range when describing the variability of a variable?

When we use the inter-quartile range compared with the range, there is a primary advantage that we neglect the possible outliers in the data by eliminating the observations below first quartile and above third quartile. We do not consider the lowest and highest observations or minimum and maximum values during the calculation of inter-quartile range. However, we use the minimum and maximum values during the calculating of range.

Can the standard deviation ever be larger than the variance? Explain.

Yes, standard deviation can be larger than the variance. In case of smaller variances (variances between 0 and 1), the standard deviations are larger than variances. (Square roots of numbers less than 1 are greater than itself.)

Can the variance ever be negative? Why or why not?

Variance can never be negative, because we know that there is zero or more variation would be exists in any data set.

Show the formula for the Coefficient of Variation and explain what it is and how it can be useful in comparing the variability of different variables.

Formula for coefficient of variation is given as below:

Coefficient of variation = C.V. = (SD/Mean)*100%

(SD = standard deviation)

Coefficient of variation is numerical coefficient used for measuring the relatively variation of the variable. It is a unit-less quantity and so it is useful for comparing the variables with different units.