Sampling error and normal distribution.

**Highlights:**

- Sampling error
- Sampling distribution
- Normal distribution
- Applications of Normal distribution.

Sampling error is a variation between the sample size and the population parameter or it is when a researcher made a mistake is selecting the wrong sample size. Sampling distribution of mean is a hypothetical distribution. It focuses on the characteristics of the mean in a given distribution. This is used in a situation where we have sub-sample.

**Assumptions that Guide the Central Limit Theorem**

- The mean is an unbiased estimate of the population mean. i.e the mean of the sample mean is equal to the population mean.
- If a population is not normal and a sample of at least 30 is drawn from the population, the mean of the population will approximate to normal
- The standard error of mean is inversely proportional to the sample size. This implies that if the sample size is large, the standard error of mean will be low and vice versa. Consequently, as the sample size increases to infinity, the error score vanishes.

Application of Measures of Central Tendency and Variability

## Sampling error and normal distribution

**Normal Distribution**

This is continuous probability distribution expressed in terms of mean and standard deviation. It focuses on or show the frequency of the data set in the distribution. The normal distribution is also called the standard normal distribution because it can also be expressed in terms of the standard z-score. Also it has a total area of 1.

**Characteristics of Normal Distribution**

- Normal distribution is unimodal. This implies that it has only one mode
- The mean = median = mode and there all lie at the centre
- It is perfectly symmetrical. i.e the left and right are equal
- Also it is asymptotic; it implies that the tail continue without touching the place, else it becomes skewed distribution if it does touch the plane.

**Z-Score**

This is a standardization technique or a basic standard score from which other standard scores are transformed. It has a mean of zero and a standard deviation of one.

Z = X – X(bar)/SD

**Example**: Assuming that a population is normally distributed and a sample of 100 students are drawn such that the mean achievement of the students is 30 and standard deviation (SD) is 12. What is the proportion of students that failed the examination if the pass mark was 40?

Solution

n =100, SD =12, X-bar = 30, X = 40

Z = X – X(bar)/SD

Z = 40 – 30/12 = 0.83

Area of Z = 0.83 = 0.2033

Therefore the proportion that failed is 0.2033 and the percentage that failed is 0.2033 x 100 = 20%

The number of students that failed is 0.2033 x 100 = 20 students.

Measures of Association – Tetrachoric and Kendall Tall Correlation

**Applications of Normal Distribution**

- The normal distribution enables us to determine measure of kurtosis and skewness.
- Normal distribution enables us to determine extent of variation or differences in performance or ability level of students.
- It creates room for some standardization or transformation with which comparison can be made among students or individual.
- Also it enables us to determine the proportion or percentage of students who passed or failed a particular examination.

Correlation Coefficient for One – Two Samples In Hypothesis Testing