A normal curve has mirror symmetry about mu, so the probability below mu - a equals the probability above mu + a.
Missing Word
symmetry
Symmetry means one side of the graph is the mirror image of the other. Every normal curve mirrors itself about its mean, so equal distances below and above mu cut off equal probabilities.
Category: Normal Curve
Question 2
The 68-95-99.7 rule for normal data is also called the empirical rule.
Missing Word
empirical
The empirical rule says that about 68, 95 and 99.7 percent of a normal distribution lie within one, two and three standard deviations of the mean. It gives quick estimates without a table.
Category: Empirical Rule
Question 3
The value of X that has a proportion 0.90 of the distribution at or below it is the 90th percentile.
Missing Word
percentile
A percentile is the value with a stated share of the distribution at or below it. For a normal variable it is found from x = mu + z sigma using the matching z-score.
Category: Normal Percentiles
Question 4
For a normal variable, P(X < x) is the area under the curve to the left of x.
Missing Word
left
Probabilities of the form P(X < x) accumulate area from the far end of the curve up to x, which is the area to the left. Tables and calculators usually report this left area.
Category: Normal Probabilities
Question 5
Since about 75 percent of a normal distribution lies below mu + 0.67 sigma, that value is the third quartile.
Missing Word
quartile
Quartiles split a distribution into four parts with equal area. The third quartile has three quarters of the area below it, and for a normal distribution it sits about 0.67 standard deviations above the mean.
Category: Normal Percentiles
Question 6
For any density curve, even a lopsided one, the value with exactly half the area on each side is the median.
Missing Word
median
The median is the 50th percentile, the point that splits the area into two equal halves. For a normal curve it equals mu, but for a skewed curve it differs from the mean.
Category: Normal Percentiles
Question 7
Because the normal distribution is continuous, P(X = 70) = 0 for any normal variable X.
Missing Word
continuous
A continuous distribution assigns probability to intervals through area, and a single point has no width. So any exact value has probability 0.
Category: Normal Probabilities
Question 8
The two ends of the normal curve, which extend without limit in each direction, are called its tails.
Missing Word
tails
The tails are the parts of a distribution far from its center. The normal curve's tails thin out toward the axis but never end, so extreme values are rare but possible.
Category: Normal Curve
Question 9
A value above the mean has a positive z-score.
Missing Word
positive
A z-score is (x - mu)/sigma, so it is positive whenever x is greater than mu. Values below the mean have negative z-scores, and the mean itself has z = 0.
Category: Z-Scores
Question 10
The x-value directly under the highest point of a density curve is its mode, even when the curve is skewed.
Missing Word
mode
The mode is the most likely region of a distribution, located at the peak of its density. For a normal curve the mode equals the mean and the median.
Category: Normal Curve
Question 11
To find a z-score, subtract the mean from x and then divide the result by sigma.
Missing Word
divide
The z-score formula is z = (x - mu)/sigma: subtract the mean, then divide by the standard deviation. The result counts standard deviations from the mean.
Category: Z-Scores
Question 12
To convert a z-score back to the original scale, multiply z by sigma and add mu.
Missing Word
multiply
Reversing the z-score formula gives x = mu + z sigma. Multiplying by sigma turns standard deviations back into original units before shifting by the mean.
Category: Z-Scores
Question 13
A normal curve has exactly one peak, so it is unimodal.
Missing Word
unimodal
A unimodal distribution has a single peak. The normal curve rises to one high point at its mean and falls away on both sides.
Category: Normal Curve
Question 14
The graph of every normal density has the shape of a bell.
Missing Word
bell
The normal curve is described as bell-shaped: high in the middle and tapering symmetrically toward both sides.
Category: Normal Curve
Question 15
In the notation X ~ N(mu, sigma), the mean and the standard deviation are the two parameters of the distribution.
Missing Word
parameters
Parameters are the fixed numbers that pin down a distribution. A normal distribution is fully specified by its mean mu, which sets the center, and its standard deviation sigma, which sets the spread.
Category: Standard Normal