Central tendency Measures and Variability

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Published on March 15, 2014

Author: PedroLMartinez

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Central Tendency Measures and Variability

Central Tendency and Variability Dr. Pedro L. Martinez

Course A Course B Course C Student Hours per week Student Hours per week Student Hours per week Joe 9 Hannah 5 Meena 6 Peter 7 Ben 6 Sonia 6 Zoey 8 Iggy 6 Kim 7 Ana 8 Louis 6 Mike 5 Jose 7 Keesha 7 Jamie 6 Lee 9 Lisa 6 Ilana 6 Joshua 8 Mark 5 Lars 5 Ravi 9 Ahmed 5 Nick 20 Kristen 8 Jenny 6 Liz 5 Loren 1 Erin 6 Kevin 6 Suppose you wanted to know how many hours students spend studying for three distinct courses. The researcher does a survey of ten students in each of the courses. On the survey, he asks the students to write down the number of hours per week they spend studying for that course. The data look like this:

Response: Strongly Disagree Disagree Neither Agree nor Disagree Agree Strongly Agree Code: 1 2 3 4 5 The Likert scale is an example of an ordinal scale. Consider five possible responses to a question, Is  our instructor is an excellent teacher?, with answers on this scale. A 5 will guarantee you an passing grade in this course!

If you worked for Pizza Hut, you might want to know what are the preferences between men and women when choosing pizza toppings. Men Preferences ______________ ______________ ______________ Women Preferences ________________ ________________ ________________

If you want to summarize the above examples you mat be able to summarize data, by using: Measures of central tendency measures: They tell you about the typical scores or Measures of variability: They tell you about how scores are spread out

We buy: 100 size 4pairs 100 size 5pairs 100 size 6 pairs 100 size 7 pairs 100 size 8 pairs 100 size 9 pairs 100 size 10 pairs 100 size 11 pairs 100 size 12 pairs 100 size 13 pairs 100 size 14 pairs 4 5 6 7 8 9 10 11 12 13 14 Shoes On Sale

9 1 0 1 1 1 2 1 3 1 46 7 854 Hint – This is the Normal Curve and the Center is the Mean

When we had the same number of all the sizes – WE DID NOT PAY ATTENTION TO THE FACT that there aren’t equal numbers of feet at each shoe size.   Conclusion – we need to know two things: 1. The Typical Score: Central Tendency 2. How are the scores spread out: Variability

Frequency MEN WOMEN

Frequency Mean 4 9 14 Shoe Size Hints: •This is the normal curve and the line in the middle is the mean. A little box on the graph represents a score.

Shoe Size of Short Jockeys Shoe Size of Tall Basketball Players 99 Nuances: Distributions can come in various other shapes: 1: Skewed with the more scores to the left or right. Positive Skew Negative Skew

2. Bimodal (also multimodal) with more than one peak Shoe Size of Men and Women combined. Which hump do you think is the male peak?

Central Tendency Mode The most common score or the score with the highest frequency. Used with Nominal, Ordinal, Measurement data Median Divides distribution in half. 50 % of scores above median & 50% below median Used with Ordinal & Measurement data. Mean Arithmetic Average. Take all the scores, add them up and divide by number of scores Mean = ΣX/N For Measurement data

Variability Range Highest score minus the Lowest Score. Used with Ordinal, Measurement data Subject to extremes Standard Deviation √[Σ(X-mean)2 /N] For Measurement data Note: the Variance equals the square of the SD The standard deviation is the most commonly used measure of variable with measurement data.

Me an St a n d a r d D e v i a t i o n

Mean St a n d a rd D e v i a t i o n 9 11 7 9 68.26% of the shoe sizes between 7 and 11 Let’s look at our shoe data If you had calculated the standard deviation of the distribution of shoe sizes and found out it was 2.0* you would know that 68.26 % of males probably have shoe sizes between 7 and 11. This 9 + 1 SD or 9 + 2

Why is the Normal Distribution so Important? Researchers sometime want to simply describe the characteristics of a population not necessarily the scores. Example: If I wanted to learn about the eating habits of college students, what would I do? How would I conduct my study?

Descriptive Statistics  Descriptive statistics are used to describe the characteristics of a sample.  Central tendency statistics tells us more about the sample. It helps u s to determine how probable are the findings. This is referred to as inferential statistics.  Inferential statistics are used to make predictions about the population based on a sample.

What is the daily calorie consumption of men vs. women?  How would inferential statistics help us to look at this research?  Null hypothesis  Relationship[ between sampling and the normal distribution  A) Representative Sampling  B) Random Sampling  C) Convenience Sampling

Curves  Skewed Curves (positive, negative)  Kurtosis-height of the curve  Peak is higher than the normal distribution then is said to be leptokurtic  When is flatter then is said to be platykurtic

Kurtosis If 68%of scores fall within the mean in a normal curve what happens in the following curves?

Normal Curve 6 When the mean, median and mode are equal, you will have a normal or bell shaped distribution of scores. Example: Scores: 7, 8, 9, 9, 10, 10, 10, 11, 11, 12, 13 Mean: 10 Median: 10 Mode: 10 Range: 6

Scores that are "bunched" at the right or high end of the scale are said to have a negative skew.  If you have data where the mean, median and mode are quite different, the scores are said to be skewed. Example: Scores: 7, 8, 9, 10, 11, 11, 12, 12, 12, 13, 13 Mean: 10.7 Median: 11 Mode: 12 Range: 6

Negatively Skewed Scores that are "bunched" at the right or high end of the scale are said to have a negative skew.

Positively Skewed In a positively skewed curve scores are bunched near the left or low end of a scale.

Question: The survey of the students in three classes showed differences in how long the students studied for each course. The mean number of hours for students in Course A was about _____, and for students in Courses B and C, the average was about ________. Does this mean Course A requires the most hours of study? Were the differences the researcher observed in study time real or just due to chance? In other words, can she generalize from the samples of students she surveyed to the whole population of students? She needs to determine the reliability and significance of her statistics.

References  Review Information at these links:  http://www.psychstat.missouristate.edu/introb ook/sbk11.htm  http://www.socialresearchmethods.net/kb/meas

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