Understanding the Normal Curve and Distribution of Men’s Heights

Consider the normal curve in the figure to the​ right, which gives relative frequencies in a distribution of​ men’s heights. The distribution has a mean of 69.2 inches and a standard deviation of 2.9 inches.

To begin, let’s understand what a normal curve represents

To begin, let’s understand what a normal curve represents. A normal curve, also known as a bell curve or Gaussian curve, is a symmetric bell-shaped curve that represents the distribution of a continuous variable. In this case, the normal curve represents the distribution of men’s heights.

The mean of the distribution is 69.2 inches. This means that the average height of men in this distribution is 69.2 inches. The mean is also the peak or the highest point of the normal curve.

The standard deviation is 2.9 inches. The standard deviation measures the average amount of variation or dispersion from the mean. In this context, it tells us how much men’s heights differ from the average height of 69.2 inches. A standard deviation of 2.9 inches means that most men’s heights fall within approximately 2.9 inches above or below the mean.

Now, let’s talk about how the normal curve can be used to analyze the relative frequencies of men’s heights. The relative frequencies on the normal curve represent the proportion of men in the given height range. The area under the curve within a specific range tells us the probability of finding a man with a height within that range.

In the normal distribution, approximately 68% of the observations fall within one standard deviation of the mean. This means that about 68% of men’s heights fall between (69.2 – 2.9) = 66.3 inches and (69.2 + 2.9) = 72.1 inches. This is represented by the area under the curve within one standard deviation of the mean.

Moreover, about 95% of the observations fall within two standard deviations of the mean. So, about 95% of men’s heights fall between (69.2 – 2.9 * 2) = 63.4 inches and (69.2 + 2.9 * 2) = 75.0 inches. This is represented by the area under the curve within two standard deviations of the mean.

Furthermore, approximately 99.7% of the observations fall within three standard deviations of the mean. Hence, about 99.7% of men’s heights fall between (69.2 – 2.9 * 3) = 60.5 inches and (69.2 + 2.9 * 3) = 77.9 inches. This is represented by the area under the curve within three standard deviations of the mean.

It is important to note that the normal distribution is a theoretical distribution and actual data may not perfectly fit the ideal curve. However, it provides a useful approximation in many real-life scenarios, including the analysis of men’s heights or any other normally distributed variable.

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