Question

the graph shows a distribution of data. what is the variance of the data? 0.0625 0.25 0.5 1.5

Explanation:

1. Recall the property of a normal - distribution for variance calculation:

• For a normal distribution, if we assume the data is normally distributed (the given graph is bell - shaped, which is a characteristic of a normal distribution), and we know that in a normal distribution, approximately 95% of the data lies within 2 standard deviations of the mean.
• If we assume the mean is at \(x = 4\) (the center of the bell - shaped curve), and we can estimate the range within which most of the data lies. Let's assume the range within which most of the data lies is from \(3.5\) to \(4.5\) (a common visual estimate for the spread of a normal distribution).
• The range within which approximately 95% of the data lies for a normal distribution is \(x=\mu\pm2\sigma\), where \(\mu\) is the mean and \(\sigma\) is the standard deviation.
• If \(\mu + 2\sigma=4.5\) and \(\mu - 2\sigma=3.5\) and \(\mu = 4\) (the center of the distribution), then from \(\mu + 2\sigma=4.5\) and substituting \(\mu = 4\), we get \(4 + 2\sigma=4.5\), so \(2\sigma=4.5 - 4=0.5\), and \(\sigma=\frac{0.5}{2}=0.25\).

1. Calculate the variance:

• The variance \(\sigma^{2}\) is related to the standard deviation \(\sigma\) by the formula \(\sigma^{2}=\sigma\times\sigma\).
• Since \(\sigma = 0.25\), then \(\sigma^{2}=0.25\times0.25 = 0.0625\).

Answer:

0.0625