Question

An IQ test has a mean of 105 and standard deviation of 20. Find the corresponding z-score for the IQ of 110.

Answer 1

Answer:

68% of people have IQs between 85 and 115 (100 +/- 15). 95% have IQs between 70 and 130 (100 +/- (2*15). 99.7% have IQs between 55 and 145 (100 +/- (3*15). We can tell a lot about a population just from knowing the mean, SD, and that scores are normally distributed

Answer 2

The Z-score for an IQ of 110 would be equal to 0.25

Given:

IQ = 110

Mean (μ) = 105

Standard Deviation (σ) = 20

To Find:

The value of the z-score for the IQ of 110.

Solution:

→ The Z-score of observation tells how much the given observation differs from the Mean.

→ The Z-score is the number of standard deviations a given observation lies above or below the mean.

→ An observation smaller than the mean has a negative Z-score while an observation greater than the mean has a positive Z-score.

→ The formula to calculate Z-score for an observation (X) is given as:

                      $Z-score=\frac{Observation-(Mean)}{S.D.}=\frac{X-(\mu)}{\sigma}$

→ The Z-score of the mean of the original distribution would be equal to zero.

→  The mean of all the Z-scores would be equal to 0. While the standard deviation of the Z-scores is always equal to 1.

→ In the given question, for an IQ of 110, the Z-score would be:

                         $Z-score=\frac{(IQ)-\mu}{\sigma} \\\\Z-score=\frac{110-105}{20}=\frac{5}{20} \\\\Z-score=\frac{1}{4}=0.25$

→ Therefore the Z-score for an IQ of 110 comes out to be equal to 0.25

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