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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman PowerPoint PPT Presentation


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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick. Chapter 3: Central Tendency. Key Terms: Don’t Forget Notecards. Central Tendency (p. 73) Mean (p. 74) Weighted Mean (p. 77) Median (p. 83) Mode (p. 87)

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COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

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Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

COURSE: JUST 3900

TIPS FOR APLIA

Developed By:

Ethan Cooper (Lead Tutor)

John Lohman

Michael Mattocks

Aubrey Urwick

Chapter 3:

Central Tendency


Key terms don t forget notecards

Key Terms: Don’t Forget Notecards

  • Central Tendency (p. 73)

    • Mean (p. 74)

  • Weighted Mean (p. 77)

  • Median (p. 83)

  • Mode (p. 87)

  • Unimodal (p. 88)

  • Bimodal (p. 88)

  • Multimodal (p. 88)

  • HINT: Review distribution shapes from Ch. 2!


More key terms think notecards

More Key Terms: Think Notecards


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 1: Find the mean for the sample of n=5 scores: 1, 8, 7, 5, 9

  • Question 2: A sample of n=6 scores has a mean of M=8. What is the value of ΣX for this sample?

  • Question 3: One sample has n=5 scores with a mean of M=4. A second sample has n=3 scores with a mean of M=10. If the two samples are combined, what is the mean for the combined sample?


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 1 Answer:

    • M =ΣX

      n

    • M =1+8+7+5+9

      5

    • M =30

      5

    • M = 6


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 2 Answer:

    • M =ΣX

      n

    • 8 =ΣX

      6

    • ΣX = 48


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 4: A sample of n=6 scores has a mean of M=40. One new score is added to the sample and the new mean is found to be M=35. What can you conclude about the value of the new score?

    • It must be greater than 40.

    • It must be less than 40.

  • Question 5: Find the values for n, ΣX, and M for the following sample:


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 4 Answer:

    • B) It must be less than 40. A score higher than 40 would have increased the mean.

  • Question 5 Answer:

    • n = 1+2+3+5+1

    • n = 12

    • Σ X = 5+4+4+3+3+3+2+2+2+2+2+1

    • Σ X = 33

    • M =33

      12

    • M = 2.75


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 6: Adding a new score to a distribution always changes the mean. True or False?

  • Question 7: A population has a mean of μ = 40.

    • If 5 points were added to every score, what would be the value for the new mean?

    • If every score were multiplied by 3, what would be the value of the new mean?


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 6 Answer:

    • False. If the score is equal to the mean, it does not change the mean.

  • Question 7 Answer:

    • The new mean would be 45. When a constant is added to every score, the same constant is added to the mean.

    • The new mean would be 120. When every score is multiplied (or divided) by a constant, the mean changes in the same way.


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 8: What is the mean of the following population?


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 9: Using the scores from question 8, fill in the following table.


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mean

  • Question 8 Answer:

    • μ = 7

  • Question 9 Answer:

Below

4

1

Below

1

Below

2

Above

4

Above


Median

Median

  • Question 10: Find the median for each distribution of scores:

    • 3, 4, 6, 7, 9, 10, 11

    • 8, 10, 11, 12, 14, 15

  • Question 11:The following is a distribution of measurements for a continuous variable. Find the precise median that divides the distribution exactly in half.


Median1

Median

  • Question 10 Answers:

    • The median is X = 7

    • The median is X = 11.5

  • Question 11 Answer:

2/3

6

1/3

5

Count 8 boxes

7

2

4

3

1

1

2

3

4

5

6

Median = 6.83


Median2

Median

  • Question 11 Explanation:

    • To find the precise median, we first observe that the distribution contains n = 16 scores. The median is the point with exactly 8 boxes on each side. Starting at the left-hand side and moving up the scale of measurement, we accumulate a total of 7 boxes when we reach a value of 6.5. We need 1 more box to reach our goal of 8 boxes (50%), but the next interval contains 3 boxes. The solution is to take a fraction of each box so that the fractions combine to give you one box. The fraction is determined by the number of boxes needed to reach 50% (numerator) and the number that exists in the interval (denominator).


Median3

Median

  • Question 11 Explanation:

    • For this example, we needed 1 out of the 3 boxes in the interval, so the fraction is 1/3. The median is the point located exactly one-third of the way through the interval. The interval for X = 7 extends from 6.5 to 7.5. The interval width is one point, so one-third of the interval corresponds to approximately 0.33 points. Starting at the bottom of the interval and moving up 0.33 points produces a value of 6.50 + 0.33 = 6.83. This is the median, with exactly 50% of the distribution (8 boxes) on each side.


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mode

  • Question 12: What is the mode(s) of the following distribution? Is the distribution unimodal or bimodal?


Course just 3900 tips for aplia developed by ethan cooper lead tutor john lohman

Mode

  • Question 12 Answers:

The modes are 2 and 8

The distribution is bimodal.

Note: While this is a bimodal distribution,

both modes have the same frequency.

Thus, there is no “minor” or “major” mode.


Selecting a measure of central tendency

Selecting a Measure of Central Tendency

  • Question 13: Which measure of central tendency is most affected if one extremely large score is added to a distribution? (mean, median, mode)

  • Question 14: Why is it usually inappropriate to compute a mean for scores measured on an ordinal scale?

  • Question 15: In a perfectly symmetrical distribution, the mean, the median, and the mode will all have the same value. (True or False)

  • Question 16: A distribution with a mean of 70 and a median of 75 is probably positively skewed. (True or False)


Selecting a measure of central tendency1

Selecting a Measure of Central Tendency

  • Question 13 Answer:

    • Mean

  • Question 14 Answer:

    • The definition of the mean is based on distances (the mean balances the distances) and ordinal scales do no measure distance.

  • Question 15 Answer:

    • False, if the distribution is bimodal.

  • Question 16 Answer:

    • False. The mean is displaced toward the tail on the left-hand side.


Central tendency and distribution shape

Central Tendency and Distribution Shape

  • Graphs make life so much easier!

Symmetrical Distributions

Negatively Skewed Distribution

Positively Skewed Distribution

Note: Median

usually falls

between mean

and mode.

Notice how the means follow the outliers


Frequently asked questions

Frequently Asked Questions

  • Interpolation

  • Real Limits

  • Median for Continuous Variables

  • Frequency Distribution

  • Cumulative Distributions

  • Weighted Mean


Frequently asked question faqs

Frequently Asked Question FAQs

  • How do I find the median for a continuous variable?


Frequently asked questions faqs

Frequently Asked Questions FAQs

  • Step 1: Count the total number of boxes.

  • Step 2: How many boxes are necessary to reach 50%?

  • Step 3: Count the necessary number of boxes starting from the left (in this case 8).

50% of 16

is 8.

14

16 boxes

10

13

16

7

9

12

11

15

4

5

6

8

3

1

2


Frequently asked questions faqs1

Frequently Asked Questions FAQs

Uh-oh!

What now?

1/3

2/3

7

1

2

3

4

5

6

6.5

7.5

  • Step 4: We need one more box to reach 8, but there are

  • three boxes over the interval spanning 6.5 – 7.5. Thus,

  • we need 1/3 of each box to reach 50%.


Frequently asked questions faqs2

Frequently Asked Questions FAQs

  • Step 5: We stopped counting when we reached seven boxes at the interval X = 6, which has an upper real limit of 6.5. We want 1/3 of the boxes in the next interval, so we add 6.5 + (1/3) = 6.83.

Median = 6.83


Frequently asked questions faqs3

Frequently Asked Questions FAQs


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