91Ó°ÊÓ

The following table gives the standard deductions and personal exemptions for persons filing with "single" status on their 2011 state income taxes in a random sample of 9 states. Calculate the mean and median for the data on standard deductions for these states. $$ \begin{array}{lcc} \hline \text { State } & \begin{array}{c} \text { Standard Deduction } \\ \text { (in dollars) } \end{array} & \begin{array}{c} \text { Personal Exemption } \\ \text { (in dollars) } \end{array} \\ \hline \text { Delaware } & 3250 & 110 \\ \text { Hawaii } & 2000 & 1040 \\ \text { Kentucky } & 2190 & 20 \\ \text { Minnesota } & 5450 & 3500 \\ \text { North Dakota } & 5700 & 3650 \\ \text { Oregon } & 1945 & 176 \\ \text { Rhode Island } & 5700 & 3650 \\ \text { Vermont } & 5700 & 3650 \\ \text { Virginia } & 3000 & 930 \\ \hline \end{array} $$

Step by step solution

01

Calculating the Mean

First, the sum of all standard deduction values is determined: \(3250 + 2000 + 2190 + 5450 + 5700 + 1945 + 5700 + 5700 + 3000 = 31935\). The mean is then calculated by dividing this sum by the number of states which is 9: \(\frac{31935}{9} = 3550\).

02

Calculating the Median

First, the standard deduction values need to be sorted from smallest to largest: \(1945, 2000, 2190, 3000, 3250, 5450, 5700, 5700, 5700\). The middle value in this set of 9 numbers is the median, which in this case is 3250.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

The mean, also known as the average, is an important concept in descriptive statistics. It's used to find the central tendency of a data set. To calculate the mean of standard deductions in our example:

• Add all the standard deduction values together: 3250, 2000, 2190, 5450, 5700, 1945, 5700, 5700, 3000.
• This sum equals 31,935.
• Then, divide this total by the number of values, which is 9 states.
• The mean is \(\frac{31935}{9} = 3550\).

The mean gives us a single value that summarizes the entire data set, making it easier to understand the overall trend.

Median Calculation

The median is another measure of central tendency. It represents the middle value in a data set. Here's how you calculate it:

• First, sort the standard deductions in ascending order: 1945, 2000, 2190, 3000, 3250, 5450, 5700, 5700, 5700.
• With 9 values, the middle position is the 5th number in this ordered list.
• The median is 3250.

Using the median is helpful because it isn't affected by extremely high or low values. This makes it a robust measure of central tendency.

Standard Deductions

Standard deductions reduce the amount of income that is subject to taxes. It's a flat amount that varies by state and can significantly affect taxable income. In our exercise, standard deductions range from 1945 to 5700 dollars among the different states.

• Standard deductions simplify the process of tax filing, as they are easier to claim than itemized deductions.
• They provide a way to lower taxable income, potentially reducing state income tax liability.

Understanding how standard deductions work is essential for effectively managing state taxes.

State Income Taxes

State income taxes are levied by individual states on income earned within their jurisdiction. These taxes vary widely from state to state and can be influenced by factors like standard deductions and personal exemptions.

• Each state sets its own tax rates and rules, affecting how much residents owe annually.
• Understanding state income tax can help in planning finances and ensuring compliance with state laws.

By knowing the deductions available, such as those considered in the exercise, individuals can better navigate their state income tax responsibilities.