Central Tendency and Variability

Dr. Pedro L. Martinez

Course A Course B Course C

Student

Hours per

weekStuden

tHours per

weekStuden

tHours per

week

Joe 9 Hannah 5 Meena 6

Peter 7 Ben 6 Sonia 6Zoey 8 Iggy 6 Kim 7Ana 8 Louis 6 Mike 5

Jose 7 Keesha 7 Jamie 6

Lee 9 Lisa 6 Ilana 6Joshua 8 Mark 5 Lars 5Ravi 9 Ahmed 5 Nick 20

Kristen 8 Jenny 6 Liz 5Loren 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

DisagreeNeither 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 orMeasures of variability: They tell you about how

scores are spread out

We buy:100 size 4pairs100 size 5pairs100 size 6 pairs100 size 7 pairs100 size 8 pairs100 size 9 pairs100 size 10 pairs100 size 11 pairs100 size 12 pairs100 size 13 pairs100 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 Tendency2. 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 withNominal, 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 scoresMean = X/N

For Measurement data

Variability

Range

Highest score minus the Lowest Score.

Used withOrdinal, Measurement

dataSubject to extremes

Standard Deviation

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.

Mean

StandardDeviation

Mean

StandardDeviation

9 11

7 9

68.26% of the shoe sizes between 7 and 11

Let’s look at our shoe dataIf 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

KurtosisIf 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. These answers are next to come. Study your chapter!

References

Review Information at these links: http://

www.psychstat.missouristate.edu/introbook/sbk11.htm

http://www.socialresearchmethods.net/kb/measure.php