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Measures of Central Tendency

Measures of Central Tendency. Quiz 1 vs. Retake. Mode. The most frequently occurring score in a data set. Can be any type of data (nominal, ordinal, etc.). Mode. Using a frequency distribution table, you can easily find the mode.

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Measures of Central Tendency

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  1. Measures of Central Tendency

  2. Quiz 1 vs. Retake

  3. Mode • The most frequently occurring score in a data set. • Can be any type of data (nominal, ordinal, etc.)

  4. Mode • Using a frequency distribution table, you can easily find the mode. • Scan the frequency column for the highest number, then see what score it is associated with.

  5. Mode • What if two scores have the highest frequency? • Then you have a “bimodal” data set, and you report both modes. • 3+ modes = multimodal • It is also possible to have a data set with no mode. (Ex: 3, 7, 6, 1) M F

  6. What is the Mode? Birthdays in: • January – March • April – June • July – September • October – December

  7. Median • The “middle score” in a data set. • The point along the score scale that separates the top 50% of the scores from the bottom 50% (the score value at the 50thpercentile). • The ___th percentile is the score at or below which ___% of the scores lie.

  8. Finding the Median (Odd Number of Scores) 2, 3, 5, 6, 7, 8, 9 6, 3, 9, 2, 5, 7, 8 Step 1: Put the scores in order. Step 2: Calculate the position of the median by using the formula (N + 1)/2 (in this case, the 4th position). Step 3: The median is the value of the score in that position (in this case, 6).

  9. Finding the Median (Even Number of Scores) 2, 3, 4, 5, 6, 7, 8, 9 4, 8, 2, 9, 3, 5, 7, 6 Step 1: Put the scores in order. Step 2: Calculate the position of the score before the median by using the formula N/2 (in this case, the 4th position). Then calculate the position of the score after the median by using the formula N/2 + 1 (in this case the 5th position). Step 3: Calculate the average of the two scores surrounding the median (in this case (5 + 6)/2 = 5.5). This is the median. 5.5

  10. Finding the Median • Remember, it is easy to find the mean with a cumulative frequency distribution table. • In this example, there are 149 scores, and (149 + 1)/2 = 75. • We look for 75 or the closest number larger than 75 in the Cum f column. • The median is the corresponding X value. In this example, it is 72 BPM. • What do you do if the median score in an even data set is between two X values? Calculate the average.

  11. Mean ( ) • The sum of scores in a distribution divided by the number of scores. • For a sample: • For a population:

  12. Mean • If you have a frequency distribution, create a new column, fX. • Multiply the X value by the frequency score for each row to get the fX value. • Sum the f column (for N) and the fX column. • Divide the sum of fX by N (the sum off).

  13. Range • The range is the difference between the highest score (HS) in the distribution and the lowest score (LS). • Ex: The range of the numbers 4, 8, 2, 9, 3, 5, 7, 6 is calculated by subtracting 2 from 9, which is equal to 7.

  14. Activity #1 • Find the mean, median, mode, and range of the following data: • 25, 93, 21, 21, 86, 91, 35, 95, 90, 22, 26, 19, 85, 99, 88, 24, 30, 31, 91, 29, 89, 37, 83, 91, 27, 91, 21, 30, 44, 97, 99, 21, 92, 84, 86, 32 • Put your name on your paper.

  15. Deviation • Make sure you understand this concept! • The difference between a score and the mean of the scores in a distribution.

  16. Mean as a Balancing Point • The sum of the differences between X and the mean will always be 0. -3 -1 +4

  17. Deviation with a Frequency Distribution • Create another column by multiplying the deviation score by the frequency score. • The sum of these scores should always be zero.

  18. Average Deviation • Since the average deviation always produces a result of 0, how can it every be useful? • By using the following formula instead, we can find the average distance (AD) of each score from the mean. • This can be a useful measure.

  19. Where are the Measures? Mean? Median? Mode?

  20. Which One Should You Use? • Mode • When you have nominal data • When you are interested in which score occurred most frequently • Median • When you have a seriously skewed distribution • When your data set is cut off at an arbitrary value • Mean • When you are interested in the population mean • When you want a reliable measure based on all of the data

  21. Why Use Measures of Central Tendency? • They are easy to remember and look at. • Ex: Retake Quiz Scores Summary Measures: • Mean: 15.4 • Median: 16.5 • Mode: 18 • σ: 4.48 (we will talk about this next week)

  22. But There are Risks • Summary data from Activity #1: • Mean: 59 • Median: 63.5 • Mode: 21 and 91 • Range: 80

  23. But Look at the Histogram Mean Mode Mode Median

  24. What’s the Take Home Message? • Always look at the data in graphical form. • You are trying to paint a picture with your data analysis, a picture that demonstrates your point but is an accurate representation of what you found. • Make sure that your summary measures are accurate representations of your data.

  25. Homework • Study for Chapter 5 Quiz • (don’t forget about range and AD from chapter 6) • Read Chapter 6 • Do Chapter 5 Homework

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