95% Confidence
The margin of error is a statistical term that indicates the range within
which the true value of a population parameter lies, based on a sample taken from that population.
It reflects the level of uncertainty or potential error in the results of a survey or experiment.
For example, if a poll reports a margin of error of Β±3%, and 50% of respondents support a particular
candidate, the true support level in the entire population is likely between 47% and 53%.
The margin of error accounts for the variability that arises from sampling only a portion of the
population rather than conducting a complete census.
The 95% confidence level is a common standard used in statistical analysis to express the degree of certainty that a sample's estimate is close to the true population parameter. When a margin of error is reported with a 95% confidence level, it means that if you were to take multiple random samples from the population and calculate the margin of error for each sample, about 95% of those margins would contain the true population value.
In practical terms, if a poll shows that 50% of respondents support a candidate with a margin of error of Β±3% at a 95% confidence level, you can be 95% confident that the true support level for the candidate in the entire population lies between 47% and 53%. The confidence level reflects the likelihood that the interval calculated from the sample will cover the true population parameter, acknowledging that there is a 5% chance that it will not.
Margin of error relative magnitude (1) and value (2)
The margin of error takes up a value based off of sample size and recorded proportion/estimated probability.
Taking an allegory for the population standard deviation (using square root of p*(1-p)), we show how the
margin of error value changes with sample size. Understandably, the more samples you take, the more
certain you grow of your estimate, and therefore, the smaller the margin of error (i.e. the smaller the
confidence interval).
The top graph of the appended html graph (shinyapps.io interactive graph) we have here allows you to
experiment with up to 3 different probability values that you may observe in a sample, and determine
how much bigger the margin of error is compared to your estimate of the proportion/probability estimate
you have of your sample. This top graph is more meaningful as it shows exactly how precise you can be with
probability values, at different magnitudes of probability. Consider the following, if I wanted to prove
that 0.05% of the population were say, green skinned, how many samples would I need to possibly even see
1 green skinned individual, if true. Now consider, how many samples do you think you would be satisfied with
taking, in order to adequately show that it is in fact, 0.05% of the world population that has green skin? How
confident would you be in your assessment with respective sample sizes taken? Let's say you got a measurement of
0.051% +/- 2.0%,is that a good measurement of your proportion to say that potentially 0.0% to 2.05% of the
population could be green skinned? On the flip side, if I was alleging that 50% of the population was
obese, and that my sampling returned a 95% CI of 50% +/- 2%, that scale makes the margin of error far
less of a problem to the precision of my sampling, no? Therefore, the ratio of the MOF, to the original
probability value was found to be of importance. Hence, we have included it as the first top graph, with
the y axis being the ratio, and the x axis referring to the number of samples.
The second plot is a far simpler one that returns the MOF for you at 95% significance.
\[\epsilon = Z \cdot \sqrt{p \cdot (1-p)/N} \]
Numbers...
The formula for the margin of error is used to quantify the uncertainty or potential error in survey results or statistical estimates. Hereβs a breakdown of each component:
π
Margin of Error - This represents the range within which the true value is expected to lie, given a certain confidence level.
π
Z-score - This is a value from the standard normal distribution corresponding to the desired confidence level. For instance, a 95% confidence level typically uses a Z-score of approximately 1.96.
π
Sample Proportion - This is the proportion of successes or positive responses in the sample. For example, if 60 out of 100 surveyed individuals favor a policy,
π
p would be 0.60.
1βp: Complement of the Sample Proportion - This represents the proportion of failures or negative responses. In the previous example,
π
N: Sample Size - This is the total number of observations or responses in the sample.
The formula calculates the margin of error by multiplying the Z-score (reflecting the desired confidence level) by the standard deviation of the sample proportion (which is
π
β
(
1
β
π
)
/
π
β
). The result gives a range within which the true population parameter is likely to fall, based on the sample data.
Determines the Range of Confidence in Estimates
The critical value is a factor used to determine the margin of error in statistical estimates, reflecting the level of confidence in the results. For a normal distribution, it's a z-score (e.g., 1.96 for a 95% confidence interval). In cases with small sample sizes or unknown variances, a t-score from the t-distribution is used instead, and it varies based on degrees of freedom. For categorical data, critical values come from the chi-square distribution, depending on confidence levels and degrees of freedom.
In our case, we are referring to the most common Gaussian/Normal distribution for determining the proportion,p ,
value. But if were working with a very small sample that you are unsure of, you would definitely rely on the t
distribution.
In execution, we are using qnorm(value), but please take note that the value you input for qnorm/qt/qchisq
is referring to the one-tail variant of the distribution. In other words, if you want to do a 95%
confidence interval for instance, you need to divide the difference from 100% i.e. (100-95)/2 = 2.5, and
deduct that from 100, to get 97.5%. This is because you are crafting the margin, which is both for values
greater than p and lower than p. The range of a confidence interval is 2*margin of error that we are calculating.
Smaller p values, greater sample sizes needed
Margin of error for probability/proportion (range 0 to 1), generally becomes larger relative to the parent value
for smaller proportion (p) values. This is understandably so, since an assertion of a proportion of a
population being of some state, is tied to the number of actual cases of that state that we can
observe in our sample sizes.
For instance, if I were to allege that only 0.05% of applicants, get into Harvard, I would need to survey
a large enough pool of applicants to reproduce that 0.05% or close to accordingly.
On the other hand, the magnitude of the margin of error, is largest for p values that are closest to 0.5,
and this is because of the nature of the quadratic contained within the margin of error formula.
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