Question

which of the following statements are true about the normal distribution? check all that apply.

data values farther from the mean are less common than data values closer to the mean.
50% of the data values lie at or above the mean.
the graph of the normal distribution is bell - shaped, with tapering tails that never actually touch the horizontal axis.
data values are spread evenly around the mean.
about 95% of all data values lie within 1 standard deviation of the mean.
the distribution is symmetric with a single peak.
the mean, median and mode are all equal and occur at the center of the distribution.

Explanation:

• Data values farther from the mean are less common than data values closer to the mean: In a normal distribution, the probability density is highest at the mean. As we move away from the mean (in either direction), the probability density decreases. So, data values farther from the mean are less likely (less common) than those closer to the mean.
• 50% of the data values lie at or above the mean: The normal distribution is symmetric about the mean. The total area under the normal - distribution curve is 1 (or 100%). Since it is symmetric, 50% of the data lies to the left (below) the mean and 50% lies to the right (at or above) the mean.
• The graph of the Normal Distribution is bell - shaped, with tapering tails that never actually touch the horizontal axis: The normal distribution has a bell - shaped curve. The tails of the normal distribution extend infinitely in both directions. Mathematically, the probability of getting a value at infinity is \(P(X = \pm\infty)=0\), but the curve never actually reaches the \(x\) - axis (horizontal axis) because for any finite \(x\) value, \(P(X=x)>0\) (in the continuous - distribution sense, we consider \(P(a < X < b)\) for intervals \((a,b)\)).
• Data values are spread evenly around the mean: Due to the symmetry of the normal distribution, for every value \(x=\mu + k\) (where \(\mu\) is the mean and \(k>0\)), there is a corresponding value \(x=\mu - k\) such that the probabilities \(P(X=\mu + k)=P(X=\mu - k)\) (in the continuous - distribution sense, \(P(\mu + k-\Delta<X<\mu + k+\Delta)=P(\mu - k-\Delta<X<\mu - k+\Delta)\) for a small \(\Delta>0\)).
• The distribution is symmetric with a single peak: The normal distribution \(N(\mu,\sigma^{2})\) has the property that \(f(\mu + x)=f(\mu - x)\) for all \(x\) (where \(f(x)\) is the probability density function), so it is symmetric about \(x = \mu\). Also, it has a single maximum (peak) at \(x=\mu\) (since \(f^{\prime}(x)=0\) gives \(x = \mu\) as the only critical point for maxima/minima and \(f^{\prime\prime}(\mu)<0\)).
• The mean, median and mode are all equal and occur at the center of the distribution: For a normal distribution \(N(\mu,\sigma^{2})\), the probability density function \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^{2}}{2\sigma^{2}}}\). The mode (value of \(x\) that maximizes \(f(x)\)) is \(x = \mu\). The median (value \(m\) such that \(P(X\leq m)=0.5\)) is also \(m=\mu\) because of symmetry. And by definition, the mean \(E(X)=\mu\).

Now, for the statement "About 95% of all data values lie within 1 standard deviation of the mean": The empirical rule (for a normal distribution) states that about 68% of the data lies within 1 standard deviation of the mean (\(P(\mu-\sigma<X<\mu + \sigma)\approx0.68\)), about 95% lies within 2 standard deviations (\(P(\mu - 2\sigma<X<\mu+2\sigma)\approx0.95\)), and about 99.7% lies within 3 standard deviations (\(P(\mu - 3\sigma<X<\mu+3\sigma)\approx0.997\)).

Answer:

• Data values farther from the mean are less common than data values closer to the mean.
• 50% of the data values lie at or above the mean.
• The graph of the Normal Distribution is bell - shaped, with tapering tails that never actually touch the horizontal axis.
• Data values are spread evenly around the mean.
• The distribution is symmetric with a single peak.
• The mean, median and mode are all equal and occur at the center of the distribution.