Statistical Analysis

Mean, Standard deviation, Standard deviation of the sample means, t-test

Reasons for using Statistics Observations result in the collection of

measureable data We can not realistically observe every

individual Statistics allows us to sample small portions

and draw conclusions about the whole Statistics measures the differences and

relationships between sets of data The larger the sample size the more

confident we can be that our results represent the whole.

We can never be 100% certain

Mean

the mean is the central tendency of the data. Example: 22.0, 25.2, 28.4, 30.0, 23.4 Mean = 25.8

Error Bars

Error bars are graphical representations of the variability of data.

Error bars can show the range or the standard deviation.

The error bars extend above and below the mean. Example: 22.0, 25.2, 28.4, 30.0, 23.4 Range: (8.0); ±4.0

Range

The range is a measure of the spread of data.

It is the difference between the largest and smallest values. If one data point is unusually large or

unusually small, it has a great effect on the range. These points are called outliers

Standard Deviation

Standard deviation is a measure of how the observations of a data set are dispersed around the mean.

Within a normal distribution, about 68% of all values lie within ±1 standard deviation from the mean

Standard Deviation Calculated using the formula:

For a sample: σn-1= √Σ(xi-x)2/(n-1) For a population: σn= √Σ(xi-x)2/n

Where xi is the individual observation x is the mean: the average of the

observations n is the number of observations

Example: 22.0, 25.2, 28.4, 30.0, 23.4 Mean = 25.8 n = 5 Try it out…

_

_

Using technology to calculate Standard Deviation

The formula is programmed into graphic and scientific calculators http://www.slideshare.net/gurustip/statist

ical-analysis-presentation

Standard Deviation of the Sample Means A sample is a group of items which are

considered all together for analysis. Items within a sample lose their individual

characteristics in the analysis. A summary statistic, e.g. sample mean, is

used to represent the information in the sample.

“Sample size” is the number of items within a group. “Number of samples” is the number of groups.

https://www.utdallas.edu/~metin/Ba3352/qch9-10.pdf

Standard Deviation of the Sample Means

When the standard deviation of each mean is unknown, the standard deviation of the sample means is calculated using: σx= √Σ(xj-x)2/m

Where xj is the mean of each sample x is the “grand mean”: the average of the

averages m = the number of means

https://www.utdallas.edu/~metin/Ba3352/qch9-10.pdf

-_ _

_

_

t-Test

t-Test is used to determine whether or not the difference between two sets of data is significant (real).

We use the mean, standard deviation, and sample size to calculate the t-value.

The calculated value is then compared to a table of t-values.

t-Test t-Tests require a null hypothesis. The null hypothesis proposes that no

statistical significance exists in a set of given observations.

The primary goal of a statistical test is to determine whether an observed data set is so different from what you would expect under the null hypothesis that you should reject the null hypothesis.

t-Test To use the table of t-values you must know

the “Degrees of Freedom” and the acceptable probability that chance alone could produce the results (p). Degrees of Freedom is determined from the sum

of the sample sizes minus 2. Statisticians like to be at least 95% certain

before drawing conclusions. At a p=0.05, there is only a 5% chance that the

results are due to chance. 95% chance that there really is a difference between the two sets of data.

t-Test calculation x = mean S = standard deviation (aka σ) n = number of values

If calculated t-value is greater than the critical value in the t-table, you can reject the null hypothesis.

Can be done using Excel