Regardless of the values specified for the parameters μ (mean) and σ (standard deviation), the shape of the Normal density curve will always be unimodal and symmetric.
True. The shape of the Normal density curve is always unimodal, meaning it has a single peak, and symmetric, meaning it is mirrored on both sides of the mean. This is true regardless of the values specified for the mean and standard deviation parameters.
No, that statement is incorrect. The shape of the normal density curve can differ depending on the values specified for the mean (μ) and standard deviation (σ).
1. Unimodal: The normal density curve is always unimodal, meaning it has a single peak. This remains true regardless of the values of μ and σ.
2. Symmetric: The normal density curve is symmetric only when μ is the center of the distribution. In other words, if the mean is set at the center, the curve will be symmetric. However, if the mean is not at the center, the curve will be skewed to the left or right. The degree of skewness also depends on the value of the standard deviation.
In summary, the normal density curve is always unimodal, but its symmetry depends on the value of the mean.
To understand why the shape of the Normal density curve is always unimodal and symmetric, we need to delve into the properties of the Normal distribution.
The Normal distribution is a continuous probability distribution that is often referred to as a bell curve due to its characteristic shape. It is defined by two parameters – the mean (μ) and the standard deviation (σ).
Unimodal means that the distribution has a single peak, with the highest point being at the mean (μ). No matter what values μ and σ take, the majority of the observations will cluster around the mean, resulting in a single peak in the distribution. This is why the Normal density curve is always unimodal.
Symmetry refers to the mirror image property of the Normal density curve. If you were to draw a vertical line at the mean (μ), the curve on one side of the line would be a reflection of the curve on the other side. This symmetry is what makes the Normal distribution special.
Mathematically, the bell-shaped curve is achieved through the formula for the probability density function (pdf) of the Normal distribution:
f(x) = (1/σ√(2π)) * exp(-(x-μ)² / (2σ²))
In this formula, (x - μ)² represents the squared distance from the mean, and σ² represents the variance. The exponentiation, division, and other operations ensure that the curve is unimodal and symmetric for any values of μ and σ.
It is important to note that while the shape of the Normal density curve is always unimodal and symmetric, its specific properties, such as the spread or the location of the peak, are determined by the values of μ and σ.