Exercises: Empirical Rule, Skewness and Outliers

The masses of a large sample of eggs are approximately normally distributed with mean $60$ g and standard deviation $4$ g.

Using the empirical rule, find the interval containing approximately $95\%$ of the masses.

State whether each distribution is skewed to the left, skewed to the right or symmetric:

1. a) A histogram with a long tail towards the higher values.

1. b) A histogram with a long tail towards the lower values.

1. c) A bell-shaped histogram with equal tails.

A data set has lower quartile $12$ and upper quartile $20$.

Find the interquartile range.

The heights of adult men in a rugby club are approximately normally distributed with mean $180$ cm and standard deviation $7$ cm.

1. a) Find the percentage of men with heights between $173$ cm and $187$ cm.

1. b) Find the percentage of men taller than $194$ cm.

1. c) In a club of $400$ men, how many would you expect to be shorter than $166$ cm?

The number of text messages sent in a day by ten students is:

$4, 6, 7, 8, 8, 9, 10, 11, 12, 45$

1. a) Find the mean and the median.

1. b) Explain which value is an outlier and how this affects the mean.

1. c) State, with a reason, whether the distribution is skewed to the left or right.

House prices in a Dublin suburb have a mean of €420,000 and a median of €365,000.

1. a) State what this suggests about the skewness of the distribution.

1. b) Explain which measure of average is more suitable to describe a typical house price.

A data set has $Q_1 = 24$, $Q_3 = 40$. A value is considered an outlier if it is more than $1.5 \times \text{IQR}$ below $Q_1$ or above $Q_3$.

1. a) Find the IQR and the limits for outliers.

1. b) The data set contains the values $5$, $18$, $44$ and $67$. State which are outliers.

1. c) Explain the effect on the mean and the median of removing the outliers.

The times taken by cyclists to complete a route are approximately normally distributed. Approximately $16\%$ of cyclists take more than $52$ minutes and approximately $2.5\%$ take less than $28$ minutes.

1. a) Use the empirical rule to find the mean and the standard deviation.

1. b) Hence find the percentage of cyclists who take between $36$ and $60$ minutes.

1. c) Comment on whether the data are consistent with a normal distribution if the longest recorded time is $100$ minutes.

Two classes sit the same test. Class A has mean $62$, median $62$ and standard deviation $5$. Class B has mean $58$, median $65$ and standard deviation $14$.

1. a) Describe the skewness of each distribution, giving a reason.

1. b) Compare the two classes, referring to a measure of centre and a measure of spread.

1. c) Class B has a single student with a score of $5$. Explain how removing this score would change the mean and the median.