7.04 Univariate data, center, and spread

Which measure of center best reflects the typical wage of a part-time employee?

Choose the measure that is appropriate for data sets with extreme values.

The median is the best measure of center that reflects the typical wage of a part-time employee due to presence of the three extreme values in the data set.

The mean is 22.9 which is not a very good estimation of the typical salary. It is pulled up by the three much larger values. This might be used to manipulate the analysis to make it look like employees are getting paid more generously than they actually are, so is not a good measure of center.

The mode is 18 which is definitely not a good measure of the typical salary. It represents the minimum salar. This might be used to justify that employees need a raise as it is much lower than the typical salary. This would not accurately reflect the employee salaries.

Calculate and interpret the chosen measure of center from part (a).

Remember that the median is the middle value of a data set ranked in order. Start by arranging the salaries in order from lowest to highest. If the total number of salaries is odd, then the median is the middle value. If the total number is even, then the median is the average of the two middle values.

18,18,18,18,19,19,19,20,20,21,21,21,22,22,22,23,23,37,38,38

There total number is even and the two middle values are both 21. This indicates that our median for the data set is 21.

The typical wage of a part-time employee at the given company is \$21\,000.00.

We can compare this to a mean of \$22\,850 and a mode of \$18\, 000.

Determine the mean number of times a speed camera issued a fine in that month. Give your answer rounded to one decimal place.

To find the mean, use the formula: \text{Mean}=\dfrac{\text{Sum of values}}{\text{Number of values}}

Add all the number of times a speed camera issued a fine and divide by the total number of cameras:

Determine the median number of times a speed camera issued a fine in that month. Give your answer rounded to one decimal place.

The median in a data set with an even number of values is the average of the two middle data values.

First half of the set: 101,\,102,\,115,\,115,\,121,\,124,\,127,\,128,\,130

Second half of the set: 143,\,146,\,162,\,162,\,163,\,178,\,183,\,194,\,977

The middle values of the set: 130,\,143

The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which statement should she make? Explain.

Choose the option which uses the larger number out of the mean and median.

The correct option is B: An investigation into speed cameras found that, on average, 182 fines were issued in one month.

The mean was heavily impacted by the value 977, so is much higher which gives the impression that a lot of money is coming from speed cameras.

A neutral and more thorough report would give some insights as to the sample size, the range, and any extreme values as they may be due to input or camera error.

Calculate the variance by hand using a spreadsheet or table. Round your answer to two decimal places.

Use the formula \sigma = \dfrac{\left(x_{1} - \mu \right)^{2}+ \left(x_{2} - \mu \right)^{2}+\ldots +\left(x_{N} - \mu \right)^{2}}{N}.

1. Calculate the population mean:

   \displaystyle \mu	\displaystyle =	\displaystyle \dfrac{33+32+32+32+31+32+32+32+32+32}{10}	Use the formula for mean
   	\displaystyle =	\displaystyle 32	Evaluate

2. Complete the table and find the sum of all \left(x-\mu\right)^2:

   \text{No. of runs} \left(x \right)	\left(x - \mu\right)	\left(x - \mu\right)^{2}
   33	1	1
   32	0	0
   32	0	0
   32	0	0
   31	-1	1
   32	0	0
   32	0	0
   32	0	0
   32	0	0
   32	0	0

   \displaystyle \left(x_{1} - \mu \right)^{2}+ \left(x_{2} - \mu \right)^{2}+\ldots +\left(x_{N} - \mu \right)^{2}	\displaystyle =	\displaystyle 1+0+0+0+1+0+0+0+0+0
   	\displaystyle =	\displaystyle 2

3. Divide by N=10 to find the variance.

   \displaystyle \sigma^2	\displaystyle =	\displaystyle \dfrac{\left(x_{1} - \mu \right)^{2}+ \left(x_{2} - \mu \right)^{2}+\ldots +\left(x_{N} - \mu \right)^{2}}{N}	Write the formula
   	\displaystyle =	\displaystyle \dfrac{2}{10}	Substitute known values
   	\displaystyle =	\displaystyle 0.2	Evaluate

The population variance is approximately \sigma^{2}=0.2.

The variance is quite small, so indicates that there is not much variation or dispersion in the data set.

Calculate his standard deviation by hand using a spreadsheet or table. Round your answer to two decimal places.

We have already calculated the variance, so to find the standard deviation, we just need to take the square root.

Find the square root of the variance:

The population standard deviation is approximately \sigma=0.45.

Use technology to find his standard deviation, rounded to two decimal places.

Use the population standard deviation function, \sigma on your calculator.

Using Statistics mode, enter each data point into your calculator, then choose One Variable Analysis.

Calculate the statistics by choosing the button with the \Sigma \text{x} symbol. Then, look for the population standard deviation, \sigma.

\sigma \approx 0.45

Notice, there was also the sample standard deviation s=0.4714 which is close to \sigma=0.4472, so would allow us to describe the spread, but not give the same precise value.

Interpret and compare the means for both schools.

The mean is also called the average and describes a typical value that balances all of the data points. Check the difference between the means provided in the summary.

We can see from the given summary that School A has a lower mean of 17.4 compared to School B that has a mean of 19.9. This shows that the typical travel time for students going to School A is shorter by 2.5 minutes compared to students going to School B.

Interpret and compare the medians for both schools.

Median is the middle value of a data set, which is not impacted by extreme values. Check the difference between the medians provided in the summary.

We can see from the given summary that School A has a lower median of 13.5 compared to School B that has a median of 19. This shows that the typical travel time for students going to School A is shorter by 5.5 minutes compared to students going to School B.

The two measures of center tell the same story that School B students have a longer journey to school than School A students. However, the median makes this conclusion more obvious because the difference is larger.

Calculate the standard deviations for both schools.

We were given the variance, so just need to use that the standard deviation is the square root of the variance.

We were given that:

So we can square root to get:

The units for the standard deviation are the same as the original context, so we can say that the standard deviation for School A is 11.92 minutes and the standard deviation for School B is 3.01 minutes.

Interpret and compare the standard deviations for both schools.

Observe the difference between the standard deviations for Schools A and B. Remember that the standard deviation quantifies how spread out the values in the data set are relative to the mean value.

The standard deviation for school A is 11.92 while it is 3.01 for school B. This suggests that the time students take to get to school A have greater variability or the values are more spread out, compared to school B. On average, there is 11.92 minutes of difference from the mean for school A, indicating higher variability. On the other hand, there is less variability or the values are closer to each other for school B,with an average difference from the mean of 3.01 minutes.

The higher standard deviation or variability for students traveling to school A suggests that they may have encountered heavier traffic conditions, frequent road closures, or other factors that contribute in increased travel time.

How many times have students in the school been to Washington, DC?

For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

This statistical question is looking at numerical data because it is asking for a count of how many times students have visited Washington, DC. It's not directly comparing two variables, so it's not bivariate.

We would be collecting discrete numerical data.

Can we accurately predict a dog's adult size based on their birth weight?

For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

This statistical question is bivariate because it involves analyzing the relationship between two variables: a dog's birth weight and its adult size. It aims to determine whether there's a predictive relationship between these two variables.

How long do people take to run 3 mile races? Is this comparable for different age groups?

For univariate data, there is one variable of interest in the study, for bivariate data we are looking for a relationship between two variables.

This statistical question is collects numerical data and compares them over categorical variables.

We can say that we are comparing sets of univariate data across different categories.

We may also say this is bivariate data, because it is looking at the relationship between two variables: the time it takes people to run a 3-mile race and their age groups. It seeks to determine whether there are differences in race times across different age groups, thus involving the comparison of two variables.

Formulate a question that could be used to explore the scenario.

To formulate a question about numerical univariate data, we need consider what variable we want to explore. In this case, we want to explore the wait times for joice replacement surgeries.

A sample question would be "What are the waiting times for joint replacement surgeries over the past year at the local hospital?"

Formulate a question that requires the use of a measure of center to explore the scenario.

To formulate a question that requires a measure of center, we need to identify which measure of center would best represent the needed data.

Waiting times for joint replacement surgeries may vary greatly. We should consider using the measure of center that is less impacted by extreme values, which is the median

A sample question would be "What is the median wait time for joint replacement surgeries across the hospitals in our region?"

Using the mean might not be ideal for this scenario because it is more sensitive to extreme values compared to the median. If there are a few unusually long waiting times, the mean will be skewed towards these values, giving a distorted representation of the typical waiting time.

The mode might not provide meaningful insight into the waiting times, because for the context of waiting times for joint replacement surgeries, it's less likely to have repeated exact waiting times due to the variability in individual cases.

Formulate a question that requires the use of a measure of dispersion to explore the scenario.

To formulate a question that requires a measure of dispersion, we need to identify which measure would best for the scenario.

A sample question would be "How do wait times for joint replacement surgeries at the local hospital vary?"

We could use any measure of spread or dispersion to answer this question.

The interquartile range would tell us about the middle half of the data, so would deemphasize those who got in very quickly or had to wait a very long time.

Standard deviation would give a good idea of the overall variation from the average, but might be harder for those without a statistical background to understand.

Using range would be straightforward and provide us a clear understanding of the spread of data by simply showing the difference between the highest and lowest values.

Determine which method would be the most appropriate and explain why.

Choose a method that is realistic, ethical and would match the question.

Acquiring secondary data would be the best option to know how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg. Many amusement parks utilize mobile apps or online platforms to provide real-time information about ride wait times. This is not only a practical approach, but would also provide wide and accurate data easily. This method respects visitors' privacy as participation is entirely optional.

Observation can be time-consuming and labor-intensive, especially for a popular attraction like a new roller coaster. It may not be feasible to continuously observe and record waiting times over an extended period. Survey can be challenging in a dynamic environment like an amusement park. Visitors may be unwilling to participate. Scientific experiment is clearly not applicable as it requires controlling the environment.

Explain how a sample could be selected to get unbiased data.

When we select a sample, we need to make sure that it is representative of the population.

We can select a sample to get unbiased data by making sure the sample is randomly selected, is big enough, and has the same characteristics as the population.

If we were doing an observation or survey, we would need to ensure that we were selecting people at a variety of times throughout the day. Like asking every 10th person who gets off the ride about how long they had to wait or tracking the wait time for every 10th person using observation.

When acquiring secondary data, we don't usually have much control over the sample as demographics are not necessarily collected.

Explain what the standard deviation of the data set might tell us.

Remember that standard deviation is the measure of how far individual values in the data set are from the mean.

In the scenario of how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg, the standard deviation tells us the variability or how dispersed the different waiting times are from the mean or average.

A higher standard deviation would mean that there are waiting times that are spread out and vary a lot throughout the day or for different seats on the ride, and a lower standard deviation tells us that the waiting times are quite consistent.

Explain what the median of the data might tell us.

Remember that the median is the middle value of the data set that is arranged in order.

In the scenario of how long people wait to ride the newest roller coaster at Busch Gardens Williamsburg, the median would tell us the usual waiting time without being affected by extremes or waiting times that are way longer or shorter than the usual.