The shape of the normal curve depends on its _
a) Mean deviation
b) Standard deviation
c) Quartile deviation
d) Correlation
The correct answer and explanation is :
Correct Answer:
b) Standard deviation
Explanation:
The shape of a normal curve, also known as a Gaussian distribution or bell curve, is determined by its standard deviation (σ). The mean (μ) determines the center of the curve, but the spread or dispersion of the data points around this mean is dictated by the standard deviation.
- Definition of a Normal Curve
A normal curve is a continuous probability distribution that is symmetric around the mean. It follows the empirical rule, where:
- 68% of data falls within one standard deviation (μ ± 1σ)
- 95% falls within two standard deviations (μ ± 2σ)
- 99.7% falls within three standard deviations (μ ± 3σ)
- Effect of Standard Deviation on the Normal Curve
- A smaller standard deviation results in a narrower and taller curve, indicating that the data points are closely packed around the mean.
- A larger standard deviation produces a wider and flatter curve, meaning the data is more spread out.
- Regardless of the standard deviation, the total area under the normal curve remains equal to 1 (100% probability).
- Why Other Options Are Incorrect:
- Mean Deviation (a): It measures dispersion but does not define the shape of the normal curve.
- Quartile Deviation (c): It focuses only on the spread of the middle 50% of data and does not determine the full curve.
- Correlation (d): Correlation measures relationships between variables, not the shape of a distribution.
Thus, the standard deviation is the key factor affecting the shape of the normal distribution.
Image:
The image illustrates how different standard deviations affect the shape of the normal curve. The narrower curve corresponds to a smaller standard deviation, while the wider curve represents a larger standard deviation. This visual representation confirms that the standard deviation directly controls the spread of the normal distribution.