Dissolved oxygen levels in mg/L of 6 different lakes in the Mumbai, Maharashtra region were recorded as follows: 5.4, 5.0, 8.1, 5.5, 6.5, and 5.5. Calculate the standard deviation, variance, coefficient of variation, skewness, and kurtosis using the given data.
Dissolved oxygen levels in mg/L of 6 different lakes in the Mumbai, Maharashtra region were recorded as follows: 5.4, 5.0, 8.1, 5.5, 6.5, and 5.5. Calculate the standard deviation, variance, coefficient of variation, skewness, and kurtosis using the given data.
Here's the breakdown of the statistical calculations for the dissolved oxygen levels in the Mumbai lakes:
Data: 5.4, 5.0, 8.1, 5.5, 6.5, 5.5
1. Mean (Average):
To calculate the mean, we sum all the values and divide by the number of values:
Mean = (5.4 + 5.0 + 8.1 + 5.5 + 6.5 + 5.5) / 6 = 36 / 6 = 6.0
2. Standard Deviation:
Standard deviation measures the spread of the data around the mean.
- Calculate the difference between each value and the mean.
- Square each of these differences.
- Find the average of these squared differences (this is the variance).
- Take the square root of the variance to get the standard deviation.
Calculations:
- (5.4 - 6.0)² = 0.36
- (5.0 - 6.0)² = 1.00
- (8.1 - 6.0)² = 4.41
- (5.5 - 6.0)² = 0.25
- (6.5 - 6.0)² = 0.25
- (5.5 - 6.0)² = 0.25
Variance = (0.36 + 1.00 + 4.41 + 0.25 + 0.25 + 0.25) / (6 - 1) = 6.52 / 5 = 1.304
Standard Deviation = √1.304 ≈ 1.142
3. Variance:
Variance is the average of the squared differences from the mean, as calculated in the standard deviation process.
Variance ≈ 1.304
4. Coefficient of Variation (CV):
The coefficient of variation is a normalized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage and is defined as the ratio of the standard deviation to the mean.
CV = (Standard Deviation / Mean) * 100
CV = (1.142 / 6.0) * 100 ≈ 19.03%
5. Skewness:
Skewness measures the asymmetry of the data distribution. A skewness of 0 indicates a perfectly symmetrical distribution.
Formula for sample skewness: ∑[(xi - mean) / s]^3 / n
- (5.4 - 6.0) / 1.142 ≈ -0.525
- (5.0 - 6.0) / 1.142 ≈ -0.876
- (8.1 - 6.0) / 1.142 ≈ 1.839
- (5.5 - 6.0) / 1.142 ≈ -0.438
- (6.5 - 6.0) / 1.142 ≈ 0.438
- (5.5 - 6.0) / 1.142 ≈ -0.438
- (-0.525)^3 ≈ -0.145
- (-0.876)^3 ≈ -0.672
- (1.839)^3 ≈ 6.213
- (-0.438)^3 ≈ -0.084
- (0.438)^3 ≈ 0.084
- (-0.438)^3 ≈ -0.084
Sum of cubed values ≈ 5.312
Skewness = 5.312 / 6 ≈ 0.885
Since the value is positive, the data is positively skewed.
6. Kurtosis:
Kurtosis measures the "tailedness" of the distribution. Higher kurtosis indicates more data concentrated in the tails (and thus fewer in the shoulders).
Formula for sample kurtosis: ∑[(xi - mean) / s]^4 / n - 3
- (5.4 - 6.0) / 1.142 ≈ -0.525
- (5.0 - 6.0) / 1.142 ≈ -0.876
- (8.1 - 6.0) / 1.142 ≈ 1.839
- (5.5 - 6.0) / 1.142 ≈ -0.438
- (6.5 - 6.0) / 1.142 ≈ 0.438
- (5.5 - 6.0) / 1.142 ≈ -0.438
- (-0.525)^4 ≈ 0.076
- (-0.876)^4 ≈ 0.590
- (1.839)^4 ≈ 11.421
- (-0.438)^4 ≈ 0.037
- (0.438)^4 ≈ 0.037
- (-0.438)^4 ≈ 0.037
Sum of the values ≈ 12.198
Kurtosis = 12.198 / 6 - 3 ≈ -0.967
Since the kurtosis is less than 0, the distribution is platykurtic (flatter tails than a normal distribution).
Summary:
- Mean: 6.0 mg/L
- Standard Deviation: ≈ 1.142 mg/L
- Variance: ≈ 1.304 (mg/L)²
- Coefficient of Variation: ≈ 19.03%
- Skewness: ≈ 0.885 (Positive Skew)
- Kurtosis: ≈ -0.967 (Platykurtic)