Analyzing Subject Variability in Marks

To determine which subject shows the highest variability of marks, we need to compare the relative variability rather than just the standard deviation, especially since the mean marks for the subjects are different. The standard deviation measures the absolute dispersion of data, but the Coefficient of Variation (CV) measures the relative dispersion with respect to the mean. It is calculated as the ratio of the standard deviation to the mean, usually expressed as a percentage.

Understanding Coefficient of Variation (CV)

The Coefficient of Variation is a statistical measure used to compare the degree of variation between data series, even if their means are drastically different. A higher CV indicates greater variability relative to the mean.

The formula for Coefficient of Variation is:

\(\text{CV} = \frac{\text{Standard Deviation (SD)}}{\text{Mean}} \times 100\%\)

Given Data for Subject Marks

The question provides the mean marks and standard deviation for 4 subjects:

Calculating Coefficient of Variation for Each Subject

Let's calculate the Coefficient of Variation for each subject:

Mathematics:

\(\text{CV}_{\text{Mathematics}} = \frac{15}{40} \times 100\%\)

\(\text{CV}_{\text{Mathematics}} = 0.375 \times 100\%\)

\(\text{CV}_{\text{Mathematics}} = 37.5\%\)

Physics:

\(\text{CV}_{\text{Physics}} = \frac{12}{28} \times 100\%\)

\(\text{CV}_{\text{Physics}} \approx 0.42857 \times 100\%\)

\(\text{CV}_{\text{Physics}} \approx 42.86\%\)

Chemistry:

\(\text{CV}_{\text{Chemistry}} = \frac{14}{38} \times 100\%\)

\(\text{CV}_{\text{Chemistry}} \approx 0.36842 \times 100\%\)

\(\text{CV}_{\text{Chemistry}} \approx 36.84\%\)

Biology:

\(\text{CV}_{\text{Biology}} = \frac{16}{36} \times 100\%\)

\(\text{CV}_{\text{Biology}} \approx 0.44444 \times 100\%\)

\(\text{CV}_{\text{Biology}} \approx 44.44\%\)

Comparing Variability using CV

Now, let's compare the calculated Coefficient of Variation for all subjects:

• Mathematics: 37.5%
• Physics: 42.86%
• Chemistry: 36.84%
• Biology: 44.44%

The subject with the highest Coefficient of Variation shows the highest relative variability in marks.

Comparing the CV values, Biology has the highest CV (44.44%).

Conclusion on Highest Variability of Marks

Based on the calculation of the Coefficient of Variation, Biology shows the highest variability of marks among the given subjects, relative to its mean marks.

Revision Table: Key Concepts on Variability

Additional Information on Data Dispersion

Understanding data dispersion or variability is crucial in statistics. Besides Standard Deviation and Coefficient of Variation, other measures of dispersion include:

• Range: The difference between the highest and lowest values in a dataset. It's simple but affected by outliers.
• Variance: The average of the squared differences from the Mean. Standard deviation is the square root of variance.
• Interquartile Range (IQR): The range of the middle 50% of the data. It's less affected by extreme values than the range.

When analyzing marks or test scores, variability can indicate how spread out the scores are. High variability might suggest a wide range of student performance, while low variability might indicate that most students performed similarly.