Chapter 3.1, Problem 1AC

Teacher Salaries

The following data from several years ago represent salaries (in dollars) from a school district in Greenwood, South Carolina.

1. First, assume you work for the school board in Greenwood and do not wish to raise taxes to increase salaries. Compute the mean, median, and mode, and decide which one would best support your position to not raise salaries.

2. Second, assume you work for the teachers’ union and want a raise for the teachers. Use the best measure of central tendency to support your position.

3. Explain how outliers can be used to support one or the other position.

4. If the salaries represented every teacher in the school district, would the averages be parameters or statistics?

5. Which measure of central tendency can be misleading when a data set contains outliers?

6. When you are comparing the measures of central tendency, does the distribution display any skewness? Explain.

To determine

The mean, median, and mode for the data of teacher’s salary and best option to support not to raise salary.

Answer to Problem 1AC

The mean for the data of teacher’s salary is 22,912.7, median is 16,650 and mode is 11,000 and to support the position of not increasing the salary, mean will be best.

Explanation of Solution

Given info:

The data for salaries (in dollars) from a school in Greenwood is,

10000	11000	11000	12,500	14,300	17,500
18,000	16,600	19,200	21,560	16,400	107,000

Calculation:

The formula to calculate the mean is,

$\begin{array}{c}\overline{\text{X}}=\frac{{\displaystyle \sum _1^n{X}_n}}{n}\\ =\frac{X_1+X_2+X_3+\mathrm{...}+X_n}{n}\end{array}$ (1)

Where,

• $X$ is the number of observers and active watchers.
• $n$ is the total number of states.

The sum of the salaries is given below:

$\begin{array}{c}{\displaystyle \sum _1^{12}X}=\left\{\begin{array}{c}10000+11000+11000+12,500+14,300+17,500+18,000+16,600+\\ 19,200+21,560+16,400+107,000\end{array}\right\}\\ =275,060\end{array}$

Substitute 275,060 for ${\displaystyle \sum _1^n{X}_n}$ and 12 for n in equation (1).

$\begin{array}{c}\overline{\text{X}}=\frac{275,060}{12}\\ =22,921.7\end{array}$

The mean for the data of salaries of teachers is $22,921.7.

Median:

Formula to calculate median when the number of terms is even,

$\text{Median}=\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{value}}{2}$

Substitute 12 for n in the above formula.

$\begin{array}{c}\text{Median}=\frac{{\left(\frac{12}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{12}{2}+1\right)}^{\text{th}}\text{value}}{2}\\ =\frac{{\left(6\right)}^{\text{th}}\text{value}+{\left(7\right)}^{\text{th}}\text{value}}{2}\\ =\frac{\text{ 16},\text{4}00+\text{16},\text{6}00}{2}\\ =16,650\end{array}$

The median for the data of salaries of teachers is $16,500.

Mode:

Mode is the value that occurs most often in the given data. In the given data of salaries of teachers, the value 11,000 occurs twice, which is more frequent than any other data.

The mode for the data of salaries of teachers is $11,000.

Best measure:

The objective is not to increase salary and out of all the measure central tendencies, mode has the lowest value of 11,000 and mean has the highest value 22,912.7.

To support the decision of not increasing salary, the central tendency with highest

value will be preferable.

Thus, the mean for the data of teacher’s salary is $22,912.7, median is $17,750 and mode is 1$1,000 and to support the position of not increasing the salary, mean will be best.

To determine

The best central tendency to support the raise in salary.

Answer to Problem 1AC

The best central tendency to support the raise in salary is Mode.

Explanation of Solution

From part (1), the mean for the data of teacher’s salary is $22,912.7, median is $17,750 and mode is $11,000.

To support the demand to raise the salary, the central tendency with highest value must be chosen.

One could say that the mode of the salaries of teachers is only $11,000.

Thus, the best central tendency to support the raise in salary is Mode.

To determine

The role of outliers in supporting the data differently.

Answer to Problem 1AC

The outliners highly affect the mean and it causes the misleading outcomes of the study.

Explanation of Solution

The outlier is an observation point that is distant from the other observations.

These observations greatly affect the mean as mean is average of all the observations. So if a value is very large as compared to other values, the mean will be very large as compared to central value median or mode.

In this case, the value $107,000 increases the average salary to $22,912, which is far from the typical teacher’s average salary.

To determine

Whether the averages be parameters or statistics.

Answer to Problem 1AC

The averages would be parameters as the salary of all the teachers is known.

Explanation of Solution

The salary of all the teachers is known, which means the data for entire population is available. There is no assuming or prediction of data.

If salaries represented every teacher in the school, the averages would be parameters, since the data from all the population is known.

To determine

The central tendency that is misleading due to outliers.

Answer to Problem 1AC

The central tendency that is misleading due to outliners is mean.

Explanation of Solution

The outlier is an observation point that is distant from the other observations.

If a value is very large as compared to other values, the mean will be very large as compared to central value median or mode as mean considers all the observations.

Thus, the central tendency that is misleading due to outliners is mean.

To determine

Whether the distribution show any skewness or not.

Answer to Problem 1AC

The distribution is positively skewed.

Explanation of Solution

The data contains observation 107,000, which is highest and very distant from the other observations.

It affects the mean and makes the mean higher than median and mode.

So, the distribution is skewed to the right.

Thus, the distribution is positively skewed.