Question

the graph shows the distribution of hand lengths, in centimeters, of the members of an orchestra. what is the variance of the data? 1 2 4 5 hand lengths (cm)

Explanation:

1. Recall the properties of a normal - distribution graph for variance determination:

• For a normal distribution, if we assume the graph is a normal distribution (the bell - shaped curve suggests so), and we know that for a normal distribution, approximately 68% of the data lies within one standard deviation (\(\sigma\)) of the mean, and approximately 95% of the data lies within two standard deviations of the mean.
• Although the data points are not given explicitly, if we assume a “standard” normal - like situation where we can estimate the spread. A common way to estimate the standard deviation from a normal - distribution graph is by looking at the inflection points. The inflection points of a normal distribution \(y = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}\) occur at \(x=\mu\pm\sigma\).
• If we assume that the interval from the start of the significant part of the curve to the peak (or vice - versa) represents approximately 2 standard deviations. Looking at the graph, if we assume the mean is around 19 and the data starts to be non - negligible around 17 and goes to 21. The range from 17 to 21 is approximately \(21 - 17=4\), and this range is approximately \(4\sigma\) (since the range from \(\mu - 2\sigma\) to \(\mu + 2\sigma\) for a normal distribution contains about 95% of the data).
• If \(4\sigma\approx4\), then \(\sigma = 1\).

1. Recall the relationship between variance and standard deviation:

• The variance \(\sigma^{2}\) is related to the standard deviation \(\sigma\) by the formula \(\text{Variance}=\sigma^{2}\).
• Since \(\sigma = 1\), then \(\text{Variance}=1^{2}=1\).

Answer:

1. Option Text