Set A - Question 1

Question Details

A fitness center wants to analyze the performance improvement of two training programs (Program X and Program Y). The performance score (out of 100) represents the improvement in clients' fitness levels after 8 weeks of training.

Data:

• Program X: 55, 60, 65, 70, 75, 60, 80
• Program Y: 58, 62, 64, 68, 70, 62, 66

Descriptive Statistics & Consistency Analysis (15 marks)

• i. Calculate the mean, median, and mode (if any) for both programs to understand the central tendency of performance improvement. (5 marks)
• ii. Compute the range, sample variance, and sample standard deviation for both programs to measure the dispersion in performance scores. (7 marks)
• iii. Based on your analysis, which training program shows higher consistency in client performance improvement? Provide a statistical justification for your conclusion using the measures computed above. (3 marks)

Additional Concept Question (5 marks)

• B. Provide difference between correlation and covariance and discuss its significance. (5 marks)

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Model Answer

A. Descriptive Statistics & Consistency Analysis (15 marks)

Preliminary Organization of Data

Both datasets comprise $n = 7$ client evaluations. We order the raw scores in ascending sequence:

• Program X (Sorted): $55, 60, 60, 65, 70, 75, 80$
• Program Y (Sorted): $58, 62, 62, 64, 66, 68, 70$

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i. Measures of Central Tendency (5 Marks)

1. Sample Mean ($\bar{x}$)

The sample mean is the arithmetic average: $\bar{x} = \frac{\sum_{i=1}^n x_i}{n}$

• Program X: $\sum x_i = 55 + 60 + 60 + 65 + 70 + 75 + 80 = 465$ $\bar{x}_X = \frac{465}{7} \approx 66.4286 \approx \mathbf{66.43}$

• Program Y: $\sum y_i = 58 + 62 + 62 + 64 + 66 + 68 + 70 = 450$ $\bar{y}_Y = \frac{450}{7} \approx 64.2857 \approx \mathbf{64.29}$

2. Median

The median is the value occupying the central position of an ordered sample. With an odd sample size $n = 7$: $\text{Rank} = \frac{n + 1}{2} = \frac{7 + 1}{2} = 4^{\text{th}} \text{ ordered observation}$

• Program X: The $4^{\text{th}}$ ordered value is $65$.
• Program Y: The $4^{\text{th}}$ ordered value is $64$.

3. Mode

The mode is the observation occurring with greatest frequency:

• Program X: The score $60$ appears twice ($\text{frequency} = 2$), while all remaining observations occur once. Hence, $\text{Mode}_X = \mathbf{60}$ (Unimodal).
• Program Y: The score $62$ appears twice ($\text{frequency} = 2$), while all remaining observations occur once. Hence, $\text{Mode}_Y = \mathbf{62}$ (Unimodal).

Central Tendency Summary

Metric	Program X	Program Y	Comparison & Analytical Insight
Mean ($\bar{x}$)	$66.43$	$64.29$	Program X achieves a slightly higher average score (+2.14 points).
Median	$65.00$	$64.00$	Medians are nearly identical, differing by only 1 point.
Mode	$60.00$	$62.00$	Both distributions are unimodal.

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ii. Measures of Dispersion (7 Marks)

1. Range

$\text{Range} = x_{\max} - x_{\min}$

• Program X: $\text{Range}_X = 80 - 55 = \mathbf{25}$
• Program Y: $\text{Range}_Y = 70 - 58 = \mathbf{12}$

2. Sample Variance ($s^2$) and Sample Standard Deviation ($s$)

Applying Bessel's correction for sample variance with degrees of freedom $(n - 1) = 7 - 1 = 6$: $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}, \qquad s = \sqrt{s^2}$

Deviation Table for Program X ($\bar{x} = 66.4286$)

Client ($i$)	Score ($x_i$)	Deviation $(x_i - \bar{x})$	Squared Deviation $(x_i - \bar{x})^2$
1	55	$-11.4286$	$130.6122$
2	60	$-6.4286$	$41.3265$
3	60	$-6.4286$	$41.3265$
4	65	$-1.4286$	$2.0408$
5	70	$+3.5714$	$12.7551$
6	75	$+8.5714$	$73.4694$
7	80	$+13.5714$	$184.1837$
Total ($\sum$)	$465$	$0.0000$	$485.7143$

Using the exact sum of squares identity: $\sum x_i^2 = 55^2 + 60^2 + 60^2 + 65^2 + 70^2 + 75^2 + 80^2 = 31375$ $\sum (x_i - \bar{x})^2 = 31375 - \frac{465^2}{7} = 31375 - \frac{216225}{7} = \frac{3400}{7} \approx 485.7143$

• Sample Variance ($s_X^2$): $s_X^2 = \frac{3400 / 7}{6} = \frac{3400}{42} = \frac{1700}{21} \approx \mathbf{80.95}$
• Sample Standard Deviation ($s_X$): $s_X = \sqrt{\frac{1700}{21}} \approx \mathbf{9.00}$

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Deviation Table for Program Y ($\bar{y} = 64.2857$)

Client ($i$)	Score ($y_i$)	Deviation $(y_i - \bar{y})$	Squared Deviation $(y_i - \bar{y})^2$
1	58	$-6.2857$	$39.5098$
2	62	$-2.2857$	$5.2244$
3	62	$-2.2857$	$5.2244$
4	64	$-0.2857$	$0.0816$
5	66	$+1.7143$	$2.9388$
6	68	$+3.7143$	$13.7959$
7	70	$+5.7143$	$32.6531$
Total ($\sum$)	$450$	$0.0000$	$99.4286$

Using the exact sum of squares identity: $\sum y_i^2 = 58^2 + 62^2 + 62^2 + 64^2 + 66^2 + 68^2 + 70^2 = 29028$ $\sum (y_i - \bar{y})^2 = 29028 - \frac{450^2}{7} = 29028 - \frac{202500}{7} = \frac{696}{7} \approx 99.4286$

• Sample Variance ($s_Y^2$): $s_Y^2 = \frac{696 / 7}{6} = \frac{696}{42} = \frac{116}{7} \approx \mathbf{16.57}$
• Sample Standard Deviation ($s_Y$): $s_Y = \sqrt{\frac{116}{7}} \approx \mathbf{4.07}$

Dispersion Summary

Metric	Program X	Program Y	Dispersion Comparison
Range	$25.00$	$12.00$	Program Y has less than half the total spread of Program X.
Sample Variance ($s^2$)	$80.95$	$16.57$	Program X variance is nearly $5\times$ that of Program Y.
Sample Standard Deviation ($s$)	$9.00$	$4.07$	Program Y scores cluster much closer to the sample mean.

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iii. Consistency Analysis & Statistical Justification (3 Marks)

1. Relative Dispersion: Coefficient of Variation ($CV$)

The Coefficient of Variation measures relative dispersion independently of scale: $CV = \left( \frac{s}{\bar{x}} \right) \times 100\%$

• Program X: $CV_X = \left( \frac{8.9974}{66.4286} \right) \times 100\% \approx \mathbf{13.54\%}$

• Program Y: $CV_Y = \left( \frac{4.0708}{64.2857} \right) \times 100\% \approx \mathbf{6.33\%}$

2. Statistical Conclusion & Justification

• Conclusion: Program Y exhibits significantly higher consistency in client performance improvement than Program X.
• Justification:
  1. Substantially Lower Dispersion: Program Y has a standard deviation of $4.07$, compared to $9.00$ for Program X.
  2. Lower Coefficient of Variation: $CV_Y (6.33\%) < CV_X (13.54\%)$, confirming that Program Y has less than half the relative variability of Program X.
  3. Analytical Takeaway: Although Program X produces a marginally higher mean (+2.14 points), its outcomes are widely volatile (scores span 55 to 80). Program Y offers predictable, reliable progress for all participants (scores tightly concentrated between 58 and 70).

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B. Difference Between Correlation and Covariance & Discussion of Significance (5 Marks)

1. Mathematical Definitions

• Sample Covariance: Quantifies the directional joint variability between two variables: $\text{Cov}(X, Y) = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{n - 1}$
• Pearson's Correlation Coefficient ($r$): The standardized covariance normalized by the product of standard deviations: $r = \frac{\text{Cov}(X, Y)}{s_X \cdot s_Y} = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^n (x_i - \bar{x})^2 \sum_{i=1}^n (y_i - \bar{y})^2}}$

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2. Comparative Matrix: Covariance vs. Correlation

Feature	Covariance	Correlation ($r$)
Primary Role	Determines the direction of joint linear movement (+ or -).	Determines both the direction and the exact strength of linear association.
Units of Measurement	Expressed in product units of variables (e.g., $\text{kg} \times \text{cm}$).	Dimensionless / Unit-free scalar ratio.
Bounded Range	Extends from $-\infty$ to $+\infty$.	Strictly bounded in $[-1, +1]$.
Scale Sensitivity	Scale-dependent: Changing measurement units changes numerical value.	Scale-invariant: Unchanged by linear transformations or unit conversions.
Strength Assessment	Cannot determine strength because magnitude depends on arbitrary units.	Magnitude directly reflects strength: $\lvert r \rvert = 1$ is perfect linearity, $0$ is none.

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3. Significance in Statistical Modeling and Data Science

1. Scale-Free Feature Comparison: Covariance cannot compare relationships across heterogeneous dimensions (e.g., income in rupees vs age in years). Correlation normalizes variables, enabling direct comparison of feature relevance.
2. Detection of Multicollinearity: In predictive regression models, high bivariate correlation ($\lvert r \rvert > 0.8$) identifies redundant predictors, alerting modelers to variance inflation and potential inversion instability.
3. Portfolio Risk Optimization: In Markowitz portfolio theory, the covariance matrix controls portfolio variance, while the correlation matrix isolates pure relational dependency between financial assets.