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

By Samuel Chukwuemeka ; B.Eng, A.A.T, M.Ed. Research on the Measures of Central Tendency and Range between Inter-related Data Sets. Data: values of a variable that are observable and measurable Mean: The average value of a data set values

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

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  1. By Samuel Chukwuemeka; B.Eng, A.A.T, M.Ed Research on the Measures of Central Tendency and Range between Inter-related Data Sets

  2. 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. Vocabulary Words

  3. 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. Let • 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 Compare these two Data Sets

  5. 5, 6, 8, 8, 9, 12 • 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) Let’s look at Data Set B

  6. 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. So, what can we conclude from these results?

  7. Let • 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) if we have a common of -1

  8. Let • 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) Again, compare the two Data sets

  9. Let • 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) if we have a ratio of 1/2

  10. 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. So, what can we conclude from these results?

  11. 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 Let’s go to Exponentiation

  12. 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 Here is something interesting that I found

  13. 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. The research should continue:

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