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.

Confidence Level and Confidence Interval

Being confident make one's self more reassured. Briefly, explanations below are for two sided confidence levels/intervals in order to simplify the idea. Saying " two sided " gives initial impression that there is something like two limits, yeah they are: upper and lower limits where the confidence interval lies in between. Example: Let's look at the population of a specific mobile phone model. Suppose we are now interested in the ' weight ' property. We found that weight property follows a normal distribution with mean value of 120 grams and a standard deviation of 1.4 grams. Weight ~ Normal (Mu, Sigma) = Normal (120, 1.4) This understanding means that majority of mobiles tested will weigh very closely to 120 grams. Yes, there should be fluctuations above and below the mean value but surely that still relatively close to mean value. Suppose a question: do you expect weights like: 121, 119.5, 122.1, 118.9? Answer: Yes , I surely expect such

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

Test of Hypothesis

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