Interpreting Standard Deviation: Videos & Practice Problems

You are evaluating year-end bonus trends at two potential employers. Companies X and Y have an average bonus of $\text{(}5,000$ with standard deviations of $\text{)}800$ and $\text{(}500$, respectively.

If you aim to receive a bonus of $\text{)}6,000$ or higher, at which company is that outcome more likely? Support your answer with statistical reasoning.

Using the Empirical Rule, estimate the number of trees in a sample of $75$ whose heights are between $18$ feet and $26$ feet. Assume the sample mean height is $22$ feet and the standard deviation is $2$ feet.

Using the Empirical Rule, determine which of the following plant heights (in inches) are unusual or very unusual. The heights for eight plants are listed as follows:

$45,52,38,49,43,56,61,40$

Assume the sample mean height is $48$ inches and the standard deviation is $6$ inches.

A wildlife biologist measures the time between sightings of a rare bird. In a sample of $n=120$ sightings, the mean time between sightings is $45.5$ minutes, with a standard deviation of $12.5$ minutes. Using Chebyshev's Theorem, at least how many of the times were between $20.5$ minutes and $70.5$ minutes?

The following are sample test scores out of $100$ for two different classes. Calculate the coefficient of variation for each class and determine which class has more relative variation.

Class A: $72$, $75$, $78$, $80$, $85$

Class B: $68$, $70$, $75$, $80$, $87$

Given the data set $2$, $4$, $7$, $9$, and $11$, calculate the mean absolute deviation.

The sample standard deviation for this data is approximately $3.56$. How does the mean absolute deviation compare? Use the formula $\text{MAD}=\frac{\Sigma \mid x-\bar{x}\mid }{n}$.

In a sample of monthly electric bills, the mean is $\mathrm{\backslash (}95$ and the standard deviation is $\mathrm{\backslash )}12$. The bills follow a bell-shaped distribution. Is a bill of $\$125$ unusual? Provide your reasoning.

In a study of laptop prices, the mean price is $\mathrm{\backslash (}1,200$ with a standard deviation of $\mathrm{\backslash )}180$. The prices follow a bell-shaped distribution. Is a laptop priced at $\$870$ unusual? Explain your reasoning.

Two datasets A and B both have mean 50. Dataset A's observations are mostly 48–52 while dataset B's observations range 30–70. Which statement about their standard deviations is correct?

Which combination of conditions must be met before applying the empirical (68–95–99.7) rule to approximate percentages within k standard deviations?

A dataset of response times shows a long right tail and median much smaller than mean. A student suggests using the empirical rule to estimate percentages within 1, 2, and 3 standard deviations. Is the empirical rule appropriate here, and why?

Using the empirical rule, what approximate percentage of observations fall between one and two standard deviations above the mean in a normal distribution?

For a normally distributed test score with mean 80, using the empirical rule, what approximate percentage of students score between 70 and 86 if the standard deviation is 4?

A factory reports that the length of widgets is normally distributed with mean 10.0 cm and standard deviation 0.5 cm. According to the empirical rule, what are the numeric bounds that contain approximately 95% of widget lengths?

A nutrition company measures package weight and finds mean 250 g and standard deviation 4 g. Using the empirical rule, what percentage of packages weigh between 242 g and 258 g?

Which of the following names refers to the same rule as the empirical rule for normally distributed data?

A teacher says "About 68% of scores lie within one standard deviation of the mean; this is the 68–95–99.7 rule." What does the name "68–95–99.7" specifically represent?

Consider a process with mean 120 units and standard deviation 3 units. Which of the following is the best interpretation of the range rule of thumb regarding a measured value of 126 units?

A bottle-filling machine has mean fill 500 mL and standard deviation 4 mL. Using the range rule of thumb, is a bottle measured at 492 mL significantly low?

If heights in a population are approximately normal with mean 170 cm and standard deviation 8 cm, what is the approximate z-score for an individual 186 cm tall and is this individual in the top ~2.5% tail according to the empirical rule?