Week 6 Statistics for comparisons Research Methods Week 6 Statistics for comparisons

Normal distribution - variations One of the measures we have is the variation from the mean. How different are behaviors or results? For example, did most people purchase the same general amount, or was there a wide range in purchases?

Normal distribution - variations Why would we care? It could tell us about our customers, our students, our patients, our co-workers. Example – The average person in our office sold 400 OMR of clothing last week. What if all 10 employees sold nearly 400 OMR? What if a few sold little and a few sold over 1000 OMR?

Normal distribution - variations cust 1 54.00 cust 2 4.00 cust 3 6.00 cust 4 48.00 cust 5 5.00 cust 6 130.00 cust 7 82.00 cust 8 259.00 Let’s take this data set. 8 people buy items in our computer store. It appears we have several people buying simple items (flash drives), others buying phones, and two buying computers.

Normal distribution - variations If we order the data by sales, we see large differences. We could calculate a mean and median and mode, but we might also be interested in the amount of variation between customers.

Measures of variation What are we looking for? If we discover purchases are very consistent, we can begin to estimate income based on customer traffic. But if sales vary widely, we have less predictive power. (Xi - X)2 n

Normal distribution - variations To determine how much variation there is, we can use the measure of Standard Deviation – (Xi - X)2 n

Normal distribution – standard deviation To determine how much variation there is, we can use the measure of Standard Deviation –

Calculating standard deviation 1- calculate the mean 2- calculate the deviations of each data point from the mean, and square the result of each. The variance is the mean of these values: 3- take the square root of the variance

Calculating standard deviation - practice 1- calculate this mean: 12, 5, 4, 8, 1, 0, 2, 4, 10, 0 2- calculate the deviations of each data point from the mean, and square the result of each. The variance is the mean of these values: 3- take the square root of the variance

Calculating standard deviation - practice 1- calculate this mean: Number of white cars leaving at each time interval Number of non-white cars leaving at each interval 2- calculate the deviations of each data point from the mean, and square the result of each. The variance is the mean of these values: 3- take the square root of the variance

What does standard deviation tell us? What if we had 2 populations with the same mean, but very different standard deviations? What would that tell us about – patient recovery times, team project times, project success data, student satisfaction?

What does standard deviation tell us? How different are the standard deviations of our two car groups? Do they look like the image below?

What does standard deviation tell us? I am doing a descriptive study about white cars leaving work. What comment do I make about their standard deviation?

Comparing two samples If the mean time white cars left was 3:45 and the mean time non-white cars left was 3:50, can we say that white car drivers leave work earlier? No. Given the amount of variation we had in the two samples, the difference might have been random. To determine if the difference was real or random, we use the t-test

The t-test The purpose is to determine if the mean of two groups is different. The two groups may be “independent” or “paired.” Independent – different groups of people or objects (our two kinds of cars) Paired sample – the same people or objects, usually before and after a treatment. (white cars before and after a memo encouraging more work)

The t-test T-test returns a value – p What is the probability the means are the same? Usually we hope for the probability to be low – less that 5% (p = .05) Example – Assume we have had a group of students take a test, and then study a website and take the test again (paired sample). If the mean score for the first test was 57%, and the mean score on the second test was 60%, we want to be sure the second mean was not just a random event, but was less than 5% the result of random variation.

The t-test Example – Assume we have had a group of students take a test, and then study a website and take the test again (paired sample). If the mean score for the first test was 57%, and the mean score on the second test was 60%, we want to be sure the second mean was not just a random event, but was less than 5% the result of random variation. Process – we calculate the two means and the standard deviation for each sample. Excel makes the process simple: Formulas – more functions – statistical functions – t.test

The t-test Process – we calculate the two means and the standard deviation for each sample. Excel makes the process simple: Formulas – more functions – statistical functions – t.test Array 1 – the results of students on the first test Array 2 – the results of students on the second test 1 tail (look for change at one end of the distribution) 1 for paired sample (2 if independent sample (like our cars)) If the number returned is <.05 we believe the two means are different (the students learned something by using the website.)

The t-test Practice – Are the means of our two car types different? EXCEL - Formulas – more functions – statistical functions – t.test Array 1 – the results of white cars Array 2 – the results of non-white cars 1 tail (look for change at one end of the distribution) 2 for independent sample If the number returned is <.05 we believe the two means are different (the drivers of white cars leave earlier)