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Research on the Measures of Central Tendency and Range between Inter-related Data Sets

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## Research on the Measures of Central Tendency and Range between Inter-related Data Sets

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By### Research on the Measures of Central Tendency and Range between Inter-related Data Sets

Samuel Chukwuemeka; B.Eng, A.A.T, M.Ed

Data: values of a variable that are observable and measurable

- Mean: The average value of a data set values
- Median: The middle value of a data set when the data set values are arranged in either ascending or descending order
- Mode: The value of a data set that occurs most frequently or the value with the highest frequency
- Range: Highest data value – Lowest data value in a data set.
- Double Midrange: Highest data value + Lowest data value in a data set.

For this study, we shall be limited to studying data sets derived from data sets by

Arithmetic operations (addition, subtraction, multiplication, and division) and by

Exponential operations

Inter-relationship between Data Sets- 4, 5, 7, 7, 8, 11 …Data Set A or A
- 5, 6, 8, 8, 9, 12 …Data Set B or B
- As you can see, Data set B is derived from Data set A by a difference of 1
- Mean of A = 42/6 = 7
- Median of A = (7+7) / 2 = 7
- Mode of A = 7
- Range of A = 11 – 4 = 7

- Mean of B = 48/6 = 8
- Median of B = (8+8) / 2 = 16/2 = 8
- Mode of B = 8
- Range of B = 12 – 5 = 7
- So, we see that the
- Mean of B = Mean of A + 1
- Median of B = Median of A + 1
- Mode of B = Mode of A + 1
- Range of B = Range of A (Range is the same)

Given two Data sets A and B, such that the elements of Data set B is derived from the elements of Data set A by a difference, d; then

- The measures of central tendency of the derived Data Set (in this case Data Set B) is equivalent to the sum of the respective measures of central tendency of the original Data set (in this case Data Set A), and the common difference.
- However, the range of the derived Data set will be equal to the range of the original Data set.

- Data set A: 4, 5, 7, 7, 8, 11
- Data set B: 3, 4, 6, 6, 7, 10
- Mean of B = 36/6 = 6
- Median of B = (6 + 6) / 2 = 12/2 = 6
- Mode of B = 6
- Range of B = 10 -3 = 7
- So, we see that the
- Mean of B = Mean of A + (-1)
- Median of B = Median of A + (-1)
- Mode of B = Mode of A + (-1)
- Range of B = Range of A (Range is the same)

- 4, 5, 7, 7, 8, 11 …Data Set A or A
- 8, 10, 14, 14, 16, 22 …Data Set B or B
- As you can see, Data set B is derived from Data set A by a ratio of 2
- Mean of B = 84/6 = 14
- Median of B = (14+14) / 2 = 14
- Mode of B = 14
- Range of B = 22 – 8 = 14
- So, we see that the
- Mean of B = Mean of A * 2
- Median of B = Median of A * 2
- Mode of B = Mode of A * 2
- Range of B = Range of A * 2 (Range is not the same)

- 4, 5, 7, 7, 8, 11 …Data Set A or A
- 2, 2.5, 3.5, 3.5, 4, 5.5 …Data Set B or B
- As you can see, Data set B is derived from Data set A by a ratio of 1/2
- Mean of B = 21/6 = 3.5
- Median of B = (3.5+3.5) / 2 = 3.5
- Mode of B = 3.5
- Range of B = 5.5 – 2 = 3.5
- So, we see that the
- Mean of B = Mean of A * 1/2
- Median of B = Median of A * 1/2
- Mode of B = Mode of A * 1/2
- Range of B = Range of A * 1/2 (Range is not the same)

Given two Data sets A and B, such that the elements of Data set B is derived from the elements of Data set A by a ratio, r; then

- The measures of central tendency of the derived Data set (in this case Data set B) is equivalent to the product of the respective measures of central tendency of the original Data set (in this case Data Set A), and the common ratio.
- Similarly, the range of the derived Data set is equal to the product of the range of the original data set and the common ratio.

What if the derived data set was derived as an exponentiation of the original data set?

- Before we continue, let’s review how we defined Double Midrange in the beginning of this presentation
- Double Midrange = Highest data value + Lowest data value
- Also, we shall limit our examples to quadratic exponents (powers of 2) only. We shall also find only the range of the derived data set.
- Take for example:
- Let
- 4, 5, 7, 7, 8, 11 …Data set A or A
- 16, 25, 49, 49, 64, 121 … Data set B or B

Range of B = Range of A * Double Midrange of A

- Let’s verify this
- 4, 5, 7, 7, 8, 11 …Data set A or A
- 16, 25, 49, 49, 64, 121 … Data set B or B
- Range of A = 11 – 4 = 7
- Double Midrange of A = 11 + 4 = 15
- Range of B = 121 – 16 = 105 OR
- Range of B = 7 * 15 = 105

What are the measures of central tendency of the derived data set if the elements of the derived data set are exponents of the elements of the original data set?

- What is the range of the derived data set if the elements of the derived data set are not square exponents of the elements of the original data set?
- Let’s also work on logarithmic, inverse, trigonometric, and other functions.

Math is a jealous subject. It does not like how you spend hours watching TV shows, football games, basketball dunks, gossiping, reading novels or other subjects, but you spend only a few minutes for it (Math).

- Make math your husband or wife! Just as you would talk to your wife everyday, study math everyday. Just as you would want to make it up with your wife if you have disagree or quarrel with her, study math and if you seem to get frustrated with it, make it up with “it” by consulting someone else (instructor, students, colleagues, internet, etc). If you call it quits with your wife, she will leave you. It’s worse with Math. If you leave it for one day, it may leave you for weeks.

Make the internet your friend and always discuss math. You can find a lot of useful math resources on the internet, including answers to your assignments! Browse for “Math” and while you spend hours on Facebook and Twitter, “mathbook” and “tweet” math

- Ask questions in class! Do not be shy or ashamed of asking questions. When you ask questions, it gives the instructor an idea of the areas he needs to address specifically or re-teach or re-explain. In addition, please be specific when you ask questions. Even if you did not understand the entire solution, ask specific questions from the beginning, and continue till the end.

We have to move to Q&A session. Any questions so far, please ask.

- You can get a copy of this presentation on my website:
- Go to www.samdom4peace.com
- Point your cursor on the link, “Tutorials”
- Click on “Math”
- Check for “Presentations” table
- Locate the presentation, Troy MathFest
- Have fun!
- Thank you for listening. Have a great day!!!

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