A friend has asked for your help understanding the below histogram of the width
ID: 3231681 • Letter: A
Question
A friend has asked for your help understanding the below histogram of the width of dragon wings observed from 190 dragons flying above Rhode Island. (a) First your friend would like to extract some summary statistics from this image. Give your friend an idea of where the mean, median, and mode(s) are. (b) Based on the histogram that your friend provided, determine where the "typical" value for this data set lies and justify your choice. (c) Gus Gaussian was also consulted, but he did not look at the histogram, instead choosing to create summary statistics. He found the standard deviation to be about 28 cm. He told your friend that about 75% of the data fell between 20 and 76. Give two statistically motivated reasons why Gus would make this statement. Then explain why this statement is misguided given the histogram that your friend has provided. (d) Being the good friend that you are, you do not want to leave your friend with only Gus' misguided notion of where most of the central data lies. Instead, provide your friend with an alternative statement about where the central 50% of the data lies. Defend your notion of spread.Explanation / Answer
The histogram above is a bell shaped curve with depression in the middle and high frequencies on either side of the middle point.
a) The mean and median of this histogram lies in the middle as this is a symmetric distribution with almost equal frequencies on either side of x=50 cm.
Thus the mean and median would be between 45 to 50 cm.
The mode of this distribution is located between 20 and 25 cm which is the highest frequency amongst all the classes.
b) The typical value of this histogram lies exactly in the middle between 45 and 50 cm as it has an unusually high frequency compared to its adjacent classes.
c) Gus found out the deviation of each of the observations from the mean and adding them up he found out the standard deviation of this distribution.
After that he found out an interval in the central position for width of wings in which 75% of the width lies. By adding up the frequencies he got the result.
Given the histogram the statement is misguided because looking at the histogram it is observed that quite high frequencies of data lies before 20 and after 76. Thus one cannot conclude form the diagram that 75% of the data lies within 20 and 76. At the same time it is not possible to conclude such details as standard deviation and confidence interval by merely looking at the data. One need to have the raw data in hand to proceed with such calculations.
d) Looking at the diagram it is seen that the central 50% of the data lies approximately between 20 and 75, the interval which Gus provided for 75% of the data. This is because high frequencies exist outside this interval which would easily account for 50% of the data.