Logic of Inferential Statistics - the goal is prediction & control. Inferential statistics are a way to calculate the probability that something will occur.

Probability - is a scientific way of stating the degree of confidence in predicting something. If we know the number of total cases and the number of "favorable" or possible times of that variable in one case, then by dividing the total number by the favorable number we know the probability of occurrences of that variable. (Example: one die has six possible cases. There is only one four on the die. Therefore the probability is 1 out of 6 in the case of one die. Divide 1 by 6, which is 1/6 or .17.) Most of the time the researcher does not know the number of cases or the favorable number and has to make probability statements to understand the amount of error possible.

Error in Sampling - (The researcher wants to know: How likely is any particular sample mean likely to occur?) The researcher wants to clarify the probability of how close the sample studied represents the whole population in which the researcher intends to generalize findings. The goal of the statistical procedure is to describe a range of probable means that would occur if multiple samples were studied. To do so the researcher begins by using descriptive statistics (mean and standard deviation) from one sample to estimate the characteristics of the population. However, this is an imperfect estimate of the population because ever sample would vary slightly from another sample. Without having to do the study over with another sample the researcher wants to calculate what the mean would be of all the means likely to occur if the study was replicated with several samples from the population. The way to calculate a sampling error is based on the statistical principle that all the means from all the samples in the population when put together would form a normal curve with its own mean (the peak) and its own standard deviation (+ or - from the mean). Thus, the sampling error is the calculation of the mean of means, and the standard deviation of this new distribution to identify the range of probable means based on the normal curve. From the mean and range of one sample, the range of population means could be calculated so that the researcher could infer how many times the population mean would be within one standard deviation of the mean of means. This tells the researcher the probability of error in the sample related to the whole population.

Óx = standard error of the mean (pronounced: sigma sub X bar)

The formula to calculate the standard error of the mean is:

Óx = Ó (This symbol represents the population standard deviation)

n (The square of n, is the square of the sample size.)

Note: The larger the sample size the smaller the standard error.

Measurement error: If the researcher uses the entire population rather than a sample from the population then sampling error is not a concern, but measurement error is. The one measure obtained is a like a sample that can be calculated using its mean and range to infer its true value.