Measures of Central Tendency and Dispersion in Descriptive Statistics

What are the requisites for good average?

From the following distribution of marks of 200 students of a college:

$$\begin{array}{c|ccccccc} \text{Marks} & 30\text{-}40 & 40\text{-}50 & 50\text{-}60 & 60\text{-}70 & 70\text{-}80 & 80\text{-}90 \ \hline \text{No. of students} & 14 & 50 & 60 & 45 & 20 & 11 \ \end{array}$$

Compute:

i. The minimum marks obtained by top 10% students.

ii. Modal marks.

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The following table shows the marks obtained by 130 students in computer science.i. Find the appropriate measure of central tendency. ii. Compute the minimum marks obtained by the top 20% of students. [5]

If 50 image of your website, 10 have black and white image, and their average scanned image occupies with 2.5 megabytes of memory. The total image occupies by the entire work 281 megabytes. Find the average occupies megabytes of those color images. [5]

Calculate Q1, D7 and P58 from the following data and interpret the results.

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The following table gives the installation time (in minutes) for hardware on 50 different computers. If the average installation time is 30.2 minutes, find missing frequencies.

$$\begin{array}{|c|ccccc|c|}\hline \text{Installation Time} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 & \text{Total} \ \text{Number of computers} & 4 & - & 10 & - & 10 & 50 \ \hline \end{array}$$

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Power Failure Duration Analysis

The length of power failure in minutes are recorded in the following table. Find $Q_3$, $D_2$ and $P_{40}$ and interpret the results.

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Measurement of computer chip's thickness (in nanometers) is recorded below. Find the mode of thickness of computer chips and interpret the result.

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Calculate Q3, D6 and P80 from the following data and interpret the results.

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Define statistics and discuss its importance in the field of computational sciences. The following are the numbers of minutes that a person had to wait for the bus to work on 20 working days: 15, 10, 2, 17, 5, 8, 3, 10, 2, 9, 5, 9, 13, 1, 10, 12, 5, 10, 8, 4. Compute mean, median, mode, standard, variance and coefficient of variation.[10]

Partition Values

What are partition values?

Partition values are values that divide a distribution into equal parts. Common partition values include quartiles (dividing into 4 parts), deciles (dividing into 10 parts), and percentiles (dividing into 100 parts).

From the following distribution of scores of 200 students of a college, compute:

i. The minimum scores obtained by top 10% students.

The top 10% corresponds to the 90th percentile. We need to find the score below which 90% of students fall.

Position of 90th percentile = $\frac{90}{100} \times 200 = 180$

Cumulative frequencies:

• 30-40: 14
• 40-50: 64
• 50-60: 124
• 60-70: 169
• 70-80: 189
• 80-90: 200

The 180th student falls in the 70-80 class.

Using the formula: $P_{90} = L + \frac{\frac{90N}{100} - CF}{f} \times h$

$$P_{90} = 70 + \frac{180 - 169}{20} \times 10 = 70 + \frac{11}{20} \times 10 = 70 + 5.5 = 75.5$$

The minimum score obtained by top 10% students is 75.5

ii. The range of middle 60% students.

The middle 60% corresponds to the 20th to 80th percentiles.

For 20th percentile: Position = $\frac{20}{100} \times 200 = 40$

The 40th student falls in the 40-50 class.

$$P_{20} = 40 + \frac{40 - 14}{50} \times 10 = 40 + \frac{26}{50} \times 10 = 40 + 5.2 = 45.2$$

For 80th percentile: Position = $\frac{80}{100} \times 200 = 160$

The 160th student falls in the 60-70 class.

$$P_{80} = 60 + \frac{160 - 124}{45} \times 10 = 60 + \frac{36}{45} \times 10 = 60 + 8 = 68$$

The range of middle 60% students is 45.2 to 68

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What aspect of summary measures of data can be explained by the measures of skewness? Kelvin Hota is the national sales manager for National Text Books. He has a sales staff of 10 who visit college professors all over the United States. Each Sunday morning he requires his sales staffs to send him a report. Listed below are the number of visits last week. Compute the five number summary. 25, 6, 10, 13, 15, 2, 18, 5, 20, 30 [5]

Write notes on any two: i. Nominal and ordinal scale ii. Kurtosis iii. Five number summary [5]

The first four moments of a distribution about x=2 are 1,2.5,5.5 and 16. Calculate the first four moments about the mean. Test the skewness and kurtosis. Interpret the results. [5]

Compute first four moments about arbitrary point 4 from following distribution and describe the characteristics of data.

$$\begin{array}{c|ccccc} X & 2 & 3 & 4 & 5 & 6 \ \hline f & 1 & 3 & 7 & 2 & 1 \ \end{array}$$

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Measures of Dispersion

Question

Two batsmen A and B made the following runs in a series of cricket matches. Who is a more consistent player, and why?

$$\begin{array}{c|ccccc} A & 10 & 0 & 56 & 80 & 24 \ B & 36 & 37 & 45 & 28 & 29 \ \end{array}$$

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What are different methods of measuring dispersion?

Sample of polythene bags from two manufacturers, A, B, are tested by a prospective buyer for bursting pressure and the results are as follows. Which set of bags has more uniform pressure? If prices are the same, which manufacturer's bags would be preferred by buyer? Use appropriate statistical tool.

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What are the roles of measure of dispersion in descriptive statistics?

Following table gives the frequency distribution of thickness of computer chips (in nanometer) manufactured by two companies.

$$\begin{array}{|c|cccccc|}\hline \text{Thickness of computer chips} & 5 & 10 & 15 & 20 & 25 & 30 \ \hline \text{Number of chips} & & & & & & \ \text{Company A} & 10 & 15 & 24 & 20 & 18 & 13 \ \text{Company B} & 12 & 18 & 20 & 22 & 24 & 4 \ \hline \end{array}$$

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Distinction Between Absolute and Relative Measures of Dispersion

Absolute Measure of Dispersion: An absolute measure expresses the dispersion in the same units as the original data. Examples include range, variance, standard deviation, and mean deviation. These measures are useful when comparing datasets with similar means and units.

Relative Measure of Dispersion: A relative measure is a dimensionless quantity expressed as a ratio or percentage of a central tendency measure (usually the mean). Examples include coefficient of variation, coefficient of range, and coefficient of quartile deviation. These measures are useful for comparing the consistency of datasets with different means or different units.

Computer Consistency Analysis

For Company A:

Mean: $\bar{x}_A = \frac{1 \times 5 + 3 \times 16 + 5 \times 13 + 7 \times 7 + 9 \times 5 + 11 \times 4}{50} = \frac{5 + 48 + 65 + 49 + 45 + 44}{50} = \frac{256}{50} = 5.12$ seconds

Variance: $\sigma_A^2 = \frac{\sum f(x - \bar{x})^2}{n} = \frac{5(1-5.12)^2 + 16(3-5.12)^2 + 13(5-5.12)^2 + 7(7-5.12)^2 + 5(9-5.12)^2 + 4(11-5.12)^2}{50} = 6.8896$

Standard Deviation: $\sigma_A = \sqrt{6.8896} = 2.624$ seconds

Coefficient of Variation: $CV_A = \frac{\sigma_A}{\bar{x}_A} \times 100 = \frac{2.624}{5.12} \times 100 = 51.25%$

For Company B:

Mean: $\bar{x}_B = \frac{1 \times 2 + 3 \times 7 + 5 \times 12 + 7 \times 19 + 9 \times 9 + 11 \times 1}{50} = \frac{2 + 21 + 60 + 133 + 81 + 11}{50} = \frac{308}{50} = 6.16$ seconds

Variance: $\sigma_B^2 = \frac{2(1-6.16)^2 + 7(3-6.16)^2 + 12(5-6.16)^2 + 19(7-6.16)^2 + 9(9-6.16)^2 + 1(11-6.16)^2}{50} = 5.8704$

Standard Deviation: $\sigma_B = \sqrt{5.8704} = 2.423$ seconds

Coefficient of Variation: $CV_B = \frac{\sigma_B}{\bar{x}_B} \times 100 = \frac{2.423}{6.16} \times 100 = 39.37%$

Conclusion: Since $CV_B (39.37%) < CV_A (51.25%)$, Company B's computers are more consistent as they have a lower coefficient of variation, indicating less relative variability in execution time.

Measurement of Dispersion and Consistency Analysis

Compute percentile coefficient of kurtosis from the following data and interpret the result.

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Define skewness and kurtosis. The first four moments about mean are 0, 14.75, 39.75 and 152.31. Compute skewness and kurtosis and interpret the results. [5]