ACTIVITY - 3

MY DATASET IS FROM THE SHEET PROVIDED BY IITM ITSELF AND THE LINK TO THAT IS PROVIDED BELOW .

Question ------> 3

Select three numerical variables, such as X1, X2, and X3 from your dataset. Create scatterplots to visualize the relationships between these variable pairs. For instance, generate scatterplots for X1 vs. X2, X2 vs. X3, and so on.Additionally, provide your insights regarding the observed relationships between these variable pairs based on the scatterplot visualizations.

Answer -------->3

X1 = Current Ratio

X2 = Quick Ratio

X3 = Cash Ratio

Summary of Observed Relationships:

X1 vs X2: Moderate negative correlation – X1 and X2 decrease and increase in opposite directions.
X2 vs X3: Strong positive correlation – X2 and X3 increase together in a consistent way.
X1 vs X3: No significant correlation – X1 and X3 appear to be independent with no linear relationship.

Overall Interpretation

The covariance values provide quantitative support for the visual patterns seen in the scatterplots:

X1 and X2: Moderate negative correlation, with a covariance of -0.618, suggesting an inverse relationship.
X2 and X3: Strong positive correlation, with a covariance of 1.406, indicating a strong direct relationship.
X1 and X3: Very weak or no correlation, with a near-zero covariance (0.018), suggesting independence.

These covariance values reinforce the relationships we observed visually, confirming that X1 and X2 are inversely related, X2 and X3 have a strong positive relationship, and X1 and X3 show no meaningful relationship.

Question ------> 4

(i) Calculate the mean and standard deviation for the three chosen numerical variables. Afterward, provide an interpretation of the results to better understand the characteristics of these variables.

(ii) Formulate a statement to find a suitable bound (both an upper bound and a lower bound) by applying Chebyshev's inequality. Subsequently, give an interpretation of this result to understand the significance of the bound in terms of the variable's behavior and variability.

Answer--------->4

Mean of X1: -0.0020532091780612037

Standard Deviation of X1: 0.9613188796765688

Mean of X2: -0.01279293549618601

Standard Deviation of X2: 1.4452640053250048

Mean of X3: -0.02123610075737699

Standard Deviation of X3: 1.2267985119354397

Chebyshev's Inequality Bounds (k=2):

X1: Lower bound = -1.9246909685311988, Upper bound = 1.9205845501750765

X2: Lower bound = -2.9033209461461955, Upper bound = 2.8777350751538235

X3: Lower bound = -2.4748331246282564, Upper bound = 2.432360923113502

Summary Interpretation

In summary:

Central Tendency: All three variables have means close to zero, suggesting a symmetric or balanced distribution around the center, likely with no significant skew.
Variability: X1X1X1 has the least variability, indicating a more compact range of values, while X2X2X2 has the highest variability, suggesting a broader spread of values. X3X3X3 has intermediate variability.

These characteristics provide further context to the scatterplot and covariance analysis:

The higher variability in X2X2X2 and X3X3X3 may contribute to their strong positive correlation.
The moderate variability of X1X1X1 and X2X2X2 aligns with their moderate negative correlation.
The lack of strong variability in X1X1X1 and X3X3X3 may help explain their very weak relationship (near-zero covariance), as their ranges don’t overlap in a meaningful way that would create a strong association.

This understanding of the mean and standard deviation of each variable helps to further characterize the data and provides context for the observed relationships between the variables.

Interpretation of Chebyshev's Inequality Bounds (with k=2k = 2k=2)

According to Chebyshev's Inequality:

At least 1−1k2=1−14=0.751 - \frac{1}{k^2} = 1 - \frac{1}{4} = 0.751−k21=1−41=0.75 or 75% of the data should fall within 2 standard deviations of the mean for each variable.

Let's examine each variable:

1. Variable X1

Bounds: [-1.925, 1.921]
Interpretation: For X1, Chebyshev’s Inequality tells us that at least 75% of the values will lie within the range from -1.925 to 1.921. This interval, centered around the mean, provides a boundary within which most of the values of X1 are expected to fall. Given that X1 has a relatively low standard deviation (0.961), this range is narrower compared to the other variables, indicating that X1 values tend to stay relatively close to the mean with lower variability.

2. Variable X2

Bounds: [-2.903, 2.878]
Interpretation: For X2, the bounds from -2.903 to 2.878 indicate that at least 75% of the values should lie within this range. Since X2 has the highest standard deviation (1.445) among the variables, this interval is wider than that for X1 and X3, reflecting higher variability in X2. The larger range suggests that X2 values are more spread out from the mean, which aligns with our previous observations about its variability.

3. Variable X3

Bounds: [-2.475, 2.432]
Interpretation: For X3, the bounds are from -2.475 to 2.432, meaning that at least 75% of the values should fall within this range. The interval is wider than that of X1 but narrower than X2, consistent with X3’s moderate standard deviation (1.227). This range suggests that X3 has a moderate spread of values around the mean, with variability higher than X1 but less than X2.

Summary of the Significance of the Bounds

Behavior and Variability: These bounds help illustrate the expected spread of each variable's values based on their standard deviations. The wider the bounds, the greater the variability and spread of the data around the mean. This aligns with our earlier interpretation:
- X1 has the smallest bounds, indicating lower variability.
- X2 has the widest bounds, indicating higher variability.
- X3 falls in between, with moderate variability.
Implication: Chebyshev’s bounds provide a non-parametric way of understanding data dispersion. They confirm that X2 is the most spread out and X1 is the most concentrated around the mean. This information can be valuable in assessing each variable's stability and consistency. Variables with narrower bounds (like X1) are more predictable, while those with wider bounds (like X2) indicate more fluctuations.

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