Data Analysis Normal Distribution

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For example, we are calculating whether the male or female sauna was more successful in attracting customers (Table 4) and whether those that use the sauna had a class or didn’t have class (Table 3). Based on that we would plan whether to upgrade the male or female sauna by performing a hypothesis test. http://study.com/academy/lesson/binomial-distribution-definition-formula-examples.html

3a. What is normal distribution?

A normal distribution, sometimes known as a bell curve because it is bell shaped and symmetrical, is a mathematical expression showing the distribution of values for continuous variables. (Textbook)

For example, the average or mean duration of most males using the sauna is 15 minutes while lesser percentages of males may linger in the sauna for a shorter time of 13minutes or a longer time of 17 minutes. An even smaller percentage of them may stay in the sauna for even shorter than 13mins or longer than 17 minutes. When the variables are constructed in the histogram and polygon, most of the values congregate around the middle so the distribution resembles the shape of a bell. The bell curve is symmetrical and the mean value (middle) separates the data on the left and right side equally.

read:http://www.statisticshowto.com/probability-and-statistics/normal-distributions/

Standard Deviation

To see if the size and height of the bell curve , standard deviation ( σ) is calculated. Standard deviation (σ) is a measure of how tightly clustered is the data surrounding the mean. When the data is spread out further, the bell curve is wider and shorter meaning the standard deviation is larger and not close to the mean. When the bell curve is steeper, the head is more narrow which means more data is packed around the mean so smaller standard deviation.

[pic 1]

Diagram 1.0 http://cfcc.edu/faculty/cmoore/Empirical_Rule.htm

Empirical Rule

Based on the above picture where the graph shows a normal distribution bell curve and we use the empirical rule after calculating the standard deviation from the data to predict and make estimates about the outcomes of the numerical digits. The empirical rule is an indicator of where the percentage of data is found in which range of the mean. Diagram 1.0 depicts 3 components to the empirical rule which is the 68% , 95% and 99.7% rule. About 68% of the values fall within 1 standard deviation meaning majority of the data are concentrated in the middle , and is shown where the left and right side 34.1% of the values. Then , to get a better approximation of where most of the values are located , most people would use the 95% rule for deducing results.

How to tell if is Normal or Non Normal Distribution?

The empirical rule is crucial in determining whether the distribution plot is normal or abnormal. To verify , compute the standard deviations and compare to the expected frequency. Points that fall outside the 3 standard deviations (99.7%) range are known as outliers. If there are many points outside the 3 standard deviation range , the distribution is an abnormal distribution.

http://math.tutorvista.com/statistics/empirical-rule.html

Applications using the Standard Normal Model

The standard normal model is important in the working society like schools and businesses to approximate probabilities in discrete or continuous probabilities. It also indicates and forecasts the low and high values in the data distribution. For example , in school , the standard normal distribution indicates the grades of the majority of students, the scores of the students that do poorly and the percentage of students that do better than the average scores. This is to estimate whether the paper set was too hard for most of the students or too easy and then the teachers would decide whether to shift the bell curve to the right to pull up the grades if too many students failed or to shift left to pull down the grades if too many students scored too well. read:http://www.statisticshowto.com/probability-and-statistics/normal-distributions/

In our application , we estimate whether whether the males and females are using the sauna more frequently and whether their duration is too long or too short. From this , we can decide if we want to upgrade the male or female sauna to entice more to visit our facilities.

However the exact probability for a continuous distribution in this case is zero as time is measured and not counted. For example, the probability of males using the sauna can be between 14.40 and 16.30 seconds or between 13 to 17 seconds but the probability of the duration of males inside the sauna for exactly 16 seconds in 0. (Textbook)

What is Sampling Distribution

A sampling distribution is defined as a probability distribution which collects all possible sample sizes (n size) from a random or given population (N) to be computed into statistic data(e.g. mean, proportion or standard deviation) to reach conclusions about a population. For example , what is the proportion of males coming into the sauna from a random sample of a day’s attendance and you calculate a mean of 22.49% after converting to percentage. Sampling distribution can be determined together with population and sample size to obtain sampling distribution of the mean.

However , sometimes we do not have the population size but we can still obtain useful information using sampling distribution and confidence interval and hypothesis testing. For example , a researcher can say he is 75% certain that the sample mean is within a certain number of standard deviations of the estimated population mean.

http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/introductory-concepts/basic-concepts/sampling-distribution/

Standard Error

The sample means differs from sample to sample depending which values of interest are selected and combined together. The Standard Error of Mean (SEM) quantifies the true mean of the population and approximates the variability between samples. http://support.minitab.com/en-us/minitab/17/topic-library/basic-statistics-and-graphs/hypothesis-tests/tests-of-means/what-is-the-standard-error-of-the-mean/

However, Standard Deviation (SD) estimates variability within a single sample. The SD quantifies scatter to see how