Question

In a normal distribution, what is the \(Z\) -score equivalent of the median? What is the \(Z\) score above which only 16 percent of the distribution lies? What percentage of the scores lie below a \(Z\) score of \(+2.0 ?\)

Short answer

In a normal distribution, the Z-score equivalent of the median is \(0\). The Z-score above which only 16% of the distribution lies is approximately \(1.0\). About 97.7% of the scores lie below a Z-score of \(+2.0\).

Step-by-step solution

Find the Z-score equivalent of the median

In a normal distribution, the mean, median, and mode are all equal. Therefore, any value equal to the mean represents the median of the distribution. Since the Z-score measures how many standard deviations a value is away from the mean, a Z-score of 0 signifies that the value is equal to the mean. So, in a normal distribution, the Z-score equivalent of the median is \(0\).

Find the Z-score above which only 16% of the distribution lies

To find the Z-score above which only 16% of the distribution lies, we must determine the Z-score that corresponds to a percentile of 84%. This is due to the fact that if 16% of the distribution lies above this Z-score, then 84% (100%-16%) of the distribution lies below it. We can consult a Z-table to find the Z-score that corresponds to the 84th percentile. Looking in a Z-table, we can see that this Z-score is approximately \(1.0\).

Find the percentage of scores below a Z-score of +2.0

If we want to determine the percentage of scores below a Z-score of +2.0, we can also consult the Z-table. Looking up a Z-score of +2.0 in the Z-table, we find that the percentile of this Z-score is approximately 97.7%. So, 97.7% of the scores lie below a Z-score of +2.0 in a normal distribution.

Key concepts

Understanding Normal Distribution

The normal distribution, often referred to as the bell curve due to its bell-shaped appearance, is a fundamental concept in statistics that illustrates how data values are dispersed around a mean. This distribution is symmetric, which means the left and right sides of the curve are mirror images of each other.

Under the normal distribution, most of the data falls close to the mean (average value), with fewer values appearing as you move away towards the extremes (tails). The exact middle of a normal distribution is the median, where 50% of the values lie below and 50% lie above that point. In the context of our exercise, since a normal distribution is symmetrical, the median corresponds exactly to the mean.

In terms of Z-scores, which standardize the number of standard deviations a data point is from the mean, the median's Z-score is always zero. This signifies that it is neither above nor below the mean, landing precisely at the center of the distribution.

The Role of Standard Deviation

Standard deviation is a measure that indicates the average distance between each data point and the mean in a set of values. It reflects the dispersion or how spread out the data is in a distribution. A low standard deviation means that the data points are close to the mean, signifying low variability, whereas a high standard deviation indicates that data are spread out across a wide range of values.

In a normal distribution, knowing the standard deviation is crucial for calculating Z-scores. Since the Z-score is the number of standard deviations a particular value lies away from the mean, it essentially transforms the data into a standard form where we can easily compare values across different scales or distributions. In the exercise, for example, a Z-score of +1.0 means the value is one standard deviation above the mean, which corresponds to being higher than roughly 84% of all data points if following a normal distribution.

Deciphering Percentiles

Percentiles are used to express the position of a particular value within a data set relative to the other values. If a value is at the 50th percentile, for instance, this means that 50% of the data in the distribution is below this value. In normal distributions, percentiles align with standard deviations which allows us to use Z-scores to find the corresponding percentile.

For example, if we are searching for the percentage of scores below a Z-score of +2.0, as in the exercise, we consult a Z-table or a standard normal distribution table which tells us the percentile. In this case, we find that approximately 97.7% of the scores lie below this Z-score, placing it in the 97.7th percentile. This is vital information for various applications, from academic assessments to professional benchmarks, as it helps understand how an individual score compares to the rest of the data set.