Central Tendency

Any population is usually best described by its central tendency and variability measures. Well, these parameters should be used to best interpreting for information on a specific property under study.

Central tendency measures the value that mostly whole data/elements are centered around or grouped at. Which means, data values tend to be like/close to this value.

The most common methodology the describe the central tendency is the mean value, also called the expected value, or average value (usually for samples). There still more methods to indicate central tendency such as mode and median.

The average (or mean) of a sample of data is just the summation divided by the data count. Whereas, the mode is the most repeated/frequent value within our data. Also the median represent the middle ordered value when sorting the data group in ascending manner.

Example: the water (or juice) bottles production is a population where we can think of the bottle capacity as an important feature/property. When we look at whole 500 ml bottles, capacity values tend to be 500 ml (which is the mean or expected value). Notice that despite all bottles capacities should be close to 500 ml, they must not be exactly 500 ml. So, some can be 499.5 ml, 500.2...etc. In general, values tend to a specific center, which is the mean value.

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The "Sample"

Anytime you aim to perform a study on the entire population, you will surely find that this task will be: Much time and/or efforts consuming as populations are normally huge . Impossible if the population is infinite (such as products). Here comes the role of taking samples. Yes! we just take a sample from the whole population, perform the study on the chosen sample, apply the results back to our population. This is the core of inferential statistics because what we do is to infer parameters/properties of the population using information from a small sample. Well, this does not mean we will obtain 100% exact accurate estimations or inferences. But to be as close as possible, sample elements should be taken randomly ! At least, being random in sample selection will mostly include the diversity of information/facts within our population.

Understanding the distribution of sample mean (x_bar)

Cool, say now we have a huge population with characteristics ( Mu, Sigma^2 ). When doing a study by sampling, we take a random sample ( size n items ) and then perform the study on the sample and conclude results back for the population. From Central Limit Theorem, we know that the sample mean will always follow a normal distribution apart from what the population distribution is, such that: x_bar ~ N (Mu, Sigma^2/n) or say: Expected (x_bar) = Mu Variance (x_bar) = Sigma^2/n Well, let's see a simple illustrating example: Suppose we have a population with mean Mu=100 . Now, we have taken a sample, and computed the sample mean, x_bar. We mostly will have x_bar near 100 but not exactly 100. OK, let take another 9 separate samples... suppose these results: First sample --> x_bar = 99.8 Second sample --> x_bar = 100.1 .. .. .. 10th sample --> x_bar = 100.3 What we see that the sample mean is usually close to real population mean, that

Conclusions of Hypothesis Testing

A general hypothesis is defined as following (eg a hypothesis on the population mean): H0: Mu = Mu0 H1: Mu != Mu0 OK, apart from we have a two or one sided hypothesis, after performing the checking and statistical tests: our conclusion should be one of the following: Rejecting the null hypothesis (H0). Failing to reject the null hypothesis (H0). The following statements for conclusions are not accurate : Accepting the null hypothesis (H0). Accepting the alternative hypothesis (H1). But why? When we fail to reject H0, it does not mean we accept H0 as a fact because we still could not prove it as a fact. But what happened is that we failed to prove it to be false. This goes like following: we have suspected new factors may affected the population mean, then we have taken all possible evidences and checking, but all checking failed to prove our suspects. As well, rejecting H0 does not mean accepting H1 as a fact. What happens in this case is we p

Test of Hypothesis

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