Question

1. Section 3.4 Measures of Position and Outliers

You Explain it – Percentiles & Quartiles:

1. The 5^thpercentile of the weight of males 36 months of age is 12.0 kg.
2. The 95^thpercentile of the length of newborn females is 53.8 cm.

One variable that is measured by online homework systems is the amount of time a student spends on homework for each section of the text. The following is a summary of the number of minutes a student spends for each section of the text for the fall 2014 semester in a college statistics class at UHWO.

Q1 = 42         Q2 = 51.5         Q3 = 72.5

3. Provide an interpretation of these results.
4. Determine and interpret the interquartile range.
5. Suppose a student spends 2 hours doing homework for a section. Is this an outlier?
6. Do you believe that the distribution of time spend doing homework is skewed or symmetric? Explain your answer.

Answer 1

Percentiles and Quartiles are nothing but the relative positions of the observations in a data set.

a. Percentile can be considered as the value below which a given percentages of values in the data set lies.Hence, 5^thpercentile of the weight of males 36 months of age is 12.0 kg would simply imply that 5% of the males (36 months of age) under study have weight less than 12 Kg.

b. 95^thpercentile of the length of newborn females is 53.8 cm would mean that length of 95% of the newborn females is less than 53.8 cm.

Given that

c. Quarter is nothing but a fourth of the whole. Thus, first quarter would be the 25th percentile, the value below which one-fourth (25%) of the values lie;  second quarter would be the 50th percentile, the value below which half (50%) of the values lie; and third quarter would be the 75th percentile, the value below which three-fourth (75%) of the values lie.

Hence, based on the given data, we may infer that,

- 25% of the values in the data set lie below 42 - 50% of the values in the data set lie below 51.5 - 75% of the values in the data set lie below 72.5

d. Interquartile range is defined as the difference between the third and the first quartile.

= 30.5

It is a measure of how spread the observations in the data set are. Here, the range of middle 50% of the data is 30.5.

e. An observation can be considered to be an outlier if it lies below or above

Here,

= 42 - 1.5 (30.5)

= -3.75

= 72.5 + 1.5 (30.5)

= 118.75

Since, the observation '2' lies within (-3.75,118.75), we may conclude that this is not an outlier.

f. A distribution is said to be symmetric if the median / second quartile lies in the middle of the first and third quartile. Here,

and

We find that the median is a lot closer to the first quartile as compared to that of the third quartile. This implies that the distribution has a longer right tail (Using box plot):

Hence, we may conclude that the distribution of time spend doing homework is skewed to the right.