 # What relationship exists between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal. Calculate the z values that correspond to the outer fences of the box plot for a normal probability distribution. Slovenujozk 2022-09-12 Answered
What relationship exists between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles? The answer depends on the true underlying probability distribution of the data. Assume for the remainder of this exercise that the distribution is normal. Calculate the z values that correspond to the outer fences of the box plot for a normal probability distribution.
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In this exercise we need to show what kind of relationship there is between the standard normal distribution and the box-plot methodology for describing distributions of data by means of quartiles.
In this part we need to find the lower outer fence of the box-plot and upper inner outer of the box-plot for the standard normal random variable Z.
We know that the lower outer fence of a box-plot is equal to:
${Q}_{L}-3\ast IQR,$
and the upper outer fence of a box-plot is equal to:
${Q}_{U}+3\ast IQR.$
We also know that the interquartile range (IQR) is equal to:
$IQR={Q}_{U}-{Q}_{L}$
From part a) we have:
${Q}_{L}={z}_{L}=-.675,$
${Q}_{U}={z}_{U}=.675$
Then we have:
IQR=.675-(-.675)=.675+.675=1.35.
According to this the lower outer fence of the box-plot and upper outer fence of the box-plot for the standard normal random variable Z is:
Lower outer fence =-.675-3*1.35=-.675-4.05=-4.725,
Upper outer fence =.675+3*1.35=.675+4.05=4.725.
Result:
Lower outer fence =-4.725; Upper outer fence =4.725

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