Question

What are the z-scores that mark the middle 95% of this distribution?

What is the raw score associated with the z-score below the mean calculated in the above question?

What is the raw score associated with the z-score above the mean calculated in the 1st question?

What is the median of this z-score distribution

Answer 1

for this question, we would first need to find α the area underneath the cure which does not lay between −z and z

α=100%−95%=1−0.95=0.05

now we also know that our Standard Normal Distribution is symmetrical, so we divide α to equally be on either side of our wanted area.

so we get:

α2=0.052=0.025

which we can use our table to find −z where Φ(−z)=0.025
thus −z=−1.96 and z=1.96

The probability of randomly selecting a score between -1.96 and +1.96 standard deviations from the mean is 95% (see Fig. 4). If there is less than a 5% chance of a raw score being selected randomly, then this is a statistically significant result.

A positive z-score indicates the raw score is higher than the mean average. For example, if a z-score is equal to +1, it is 1 standard deviation above the mean.

A negative z-score reveals the raw score is below the mean average. For example, if a z-score is equal to -2, it is 2 standard deviations below the mean.

A z-score can be placed on a normal distribution curve. Z-scores range from -3 standard deviations (which would fall to the far left of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve). In order to use a z-score, you need to know the mean μ and also the population standard deviation σ.

Z-scores are a way to compare results to a “normal” population. Results from tests or surveys have thousands of possible results and units; those results can often seem meaningless. For example, knowing that someone’s weight is 150 pounds might be good information, but if you want to compare it to the “average” person’s weight, looking at a vast table of data can be overwhelming (especially if some weights are recorded in kilograms). A z-score can tell you where that person’s weight is compared to the average population’s mean weight.

The modified z score is a standardized score that measures outlier strength or how much a particular score differs from the typical score. Using standard deviation units, it approximates the difference of the score from the median.

The modified z score might be more robust than the standard z score because it relies on the median for calculating the z score. It is less influenced by outliers when compared to the standard z score.

The standard z score is calculated by dividing the difference from the mean by the standard deviation. The modified z score is calculated from the mean absolute deviation (MeanAD) or median absolute deviation (MAD). These values must be multiplied by a constant to approximate the standard deviation.

Depending on the value of MAD, the modified z score is calculated in one of two ways:

• If MAD does equal 0. Subtract the median from the score and divide by 1.253314*MeanAD. 1.253314*MeanAD approximately equals the standard deviation: (X-MED)/(1.253314*MeanAD).

• If MAD does not equal 0. Subtract the median from the score and divide by 1.486*MAD: (X-MED)/(1.486*MAD). 1.486*MAD approximately equals the standard deviation.
• The graph given below gives a better understanding
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