Standard Normal Distribution Discussion and Inferential Statistics Article Critique
Introduction
Standard normal distribution in statistics refers to data that is normally distributed in a
symmetrical system. Normality is mostly a descriptive issue. The behaviors that are described as
normal in human beings entail the major proportions that are found in a particular region. The
central tendency measures have values that are similar while the value of the standard deviation
is about 1/6 of the data range. In all normal distributions the characteristics are the same
(Krishnamoorthy, 2006).
The mean of the population compared to the standard deviation that’s below the mean is
always 34.13% of the total area under the normal curve. In a normal distribution, the region -1ϭ
to +1ϭ makes up 68.26% of the area under normal distribution (Lyon, 2014). For example, if a
scale has µ= 100 while ϭ = 20 then 68% of the total population would have an average
intelligence that range between 80 to 120 in scores while in other area like in motivation levels
where µ= 40 while ϭ = 10 then the average achievement would be from 30 to 50. In measures of
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anxiety where µ= 20 while ϭ = 10, on average 68% of the population would roughly score within
a range of 10 to 30 (Keren & Corbis, n, d).
In areas where µ to +2 ϭ is approximately 47.72% of the total population then an average
95% would be achieved i.e. 2* 47.72 of the general population would certainly have an average
score of about 70 to 130 (100 Plus 30). In areas where plus 3ϭ (49.87%) to -3then it would
include all the people in all the normal distributions.
To conclude, on average about 68% of most values are drawn from normal distribution and are
within a single standard deviation away from the distribution mean. About 95% of all the values
must lie between two standard deviations while 99.7% of all the values must be at least within
the three standard deviations.
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2) Inferential critique in statistics draws a lot of issues from empirical research studies that have
been conducted and have to be presented in a logical sequence. The following article highlights
several inferential statistical models that have been applied successfully in a lot of studies.
The Z scores are the individual scores in a normal statistical distribution and the formula for
converting the scores is known as the z transformation (Fraile & Corbis, n, d).
The formula is; z = x-M/s where;
Z = Transformed Score
X = Original Score
M= Mean of the initial Scores (Before Transformation)
S = Standard Deviation 9Before the Transformation)
The major limitation of the Z scores and tests is that, the z-test has a need for a value that
represents the population standard margin of error of the ϭ M (mean). The population standard
error is very difficult to calculate especially when the researcher has limited access to population
data.
The z-tests allow limited comparison. Only one type of comparison is feasible with the z-tests I a
sample population. Where two groups of researchers feel inclined to share comparisons then z-
test cannot be of much use as it cannot be applied in such a circumstance.
The t-test estimates the standard error (SE m) of the mean using SE m instead of ϭ M (mean).
Fraken & Corbis, n, d). The t-test formula is t = M – µM/SE m where;
T = Calculated t value
M = Sample Mean
µM = Mean of all the distribution of the sample means.
SE m = Estimated mean Standard error
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The limitations of the t-Test are related with the assumptions that all the samples taken under t-
test are independent. The assumptions of the t-Test also take it into account that all the samples
taken have been randomly selected.
The two samples under t-Test must have similar or equivalent homoscedasticity or variability.
The t-Test must also have a least interval scale to ensure accuracy of the processed information.
The two statistical inferences are applied in different fields and circumstances. The Z score is
applied in Z-test while the t score is mostly in t-Test and it’s used mainly for information on
population. The Z test can be applied in situations where the standard deviation as well as the
population mean is known. T-test is more accurate on samples that are more than 30 entries (n
> 30). The Z tests are difficult to apply when calculating probabilities.
Standard normal distribution in statistics refers to data that is normally distributed in a
symmetrical system and its significance in statistical inferences is enormous (Lyon, 2014).
The major problems with normal distribution occurs when there is a need to compare an
individual’s performance or achievements across different multiple normal distributions
measures.
To conclude, inferential statistics, the data samples are critical as they reveal the accuracy of the
represented population. The degree of accuracy in inferential statistics largely depends on the
law of large numbers (U.S. Department of Health and Human Services, 2008). The principle
asserts that errors diminish as a result of large samples (Krishnamoorthy, 2006). But the larger
the sample the more expensive is the exercise of data collection. According to Nair, Collins &
Napolitano (2013) the study on the effect of smoking and physical activity in relation to the urge
to smoke and body weight control, a larger sample may have provided more conclusive results.
But larger samples require a lot of resources and time to gather the required amount of research
data. The analysis and evaluation of data also would require more time and also resources. The
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process of deciding the right method to apply in each particular situation also requires sound and
skilful judgment.
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References
Fraile, V. & Corbis (n, d) Chapter 4, Applying the Z to Groups
Fraken, O. & Corbis (n, d) Chapter 5, The t-Test
Keren, S. & Corbis (n, d) Chapter 3, The Standard Normal Distribution and Z- Scores
Krishnamoorthy, K. (2006). Handbook of Statistical Distributions with Applications, Chapman
& Hall/CRC.
Lyon, A. (2014). Why are Normal Distributions Normal? The British Journal for the Philosophy
of Science.
Nair, S.U., Collins, N.B. & Napolitano, M.A. (2013) Differential Effects of a Body Image
Exposure Session on Smoking Urge between Physically Active Sedentary Female Smokers,
Psychology of Addictive behaviors, Vol.27. No. 1, pg. 322 – 327.
U.S. Department of Health and Human Services (2008), 2008 Physical Activity Guidelines for
Americans,
