Normal Distribution: Estimating Mean and Standard Deviation
What is the best way to estimate the mean and standard deviation of a Normal distribution?
Choose the correct option:
A. The mean is at the peak and standard deviation is the distance to the inflection points
B. The median and mode determine the standard deviation
C. The mean is the distance from the median and the standard deviation is at the peak
D. The mode is the peak and mean defines the shape of the curve
Answer:
The mean of a Normal density curve is at the peak, while the standard deviation is the distance from the mean to the inflection points. A Normal distribution is symmetric with the mean and standard deviation defining its shape.
Explanation: In a Normal density curve, the mean is represented by the peak, or the highest point of the curve, which is also the line of symmetry for the distribution. The standard deviation is the distance from the mean to the inflection point of the curve, where it changes from being concave up to concave down or vice versa. Without the actual figure, it's challenging to provide an accurate estimation.
The properties of a Normal distribution curve include:
- It is symmetric
- The mean, median, and mode occur at the highest point
- It is defined by the mean and the standard deviation
The inflection points of the curve are one standard deviation away from the mean on either side. About 68% of data will fall within one standard deviation of the mean, 95% within two, and 99.7% within three in a Normal distribution.
Therefore, the mean and standard deviation in a Normal distribution could best be estimated by identifying the peak (mean) and calculating the distance from this peak to the inflection point (standard deviation).