Margin of error – Definition, Formula + Application
By busayo.longe
| Last updated:
Jul 27
|
11 min read
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A margin of error is a statistical measurement that accounts for the difference between actual and projected results in a random survey sample. In simpler terms, the margin of error allows you to gauge the level of unpredictability in data and research outcomes.
For example, let’s say a researcher gives the mean estimate for a survey as 50 with a margin of error of ±5. This means that the actual mean, when calculated, could be any value from 45–55.
The margin of error of a data set directly influences the confidence level of the investigator in the research outcomes. Typically, if your research outcome or survey results have a higher margin of error, it means that the data sets might be unreliable. On the other hand, a lower margin of error indicates that the results truly represent the research population.
Margin of error accounts for any disparities between results from the research sample and the target population. Typically, the higher the margin of error, the greater the difference between actual and estimated outcomes, further suggesting that the survey results are less likely to be true for the entire population.
A low margin of error shows that the sample population closely represents the population of interest in a systematic investigation.
When dealing with a large population of interest, it is impossible to collect data from every variable. So, the researcher has to select a sample population that represents all the subgroups in the original population of interest.
If the sample population excludes some of the groups, then the final results of the poll, survey, or research will not reflect the actual outcome of the original population.
When you know the margin of error of your survey and the value of your observed score, you can easily determine your data set’s confidence interval. Confidence interval is a statistical range of the possible values of a parameter. In contrast, confidence level measures the accuracy rate of your research outcomes of survey results based on the available data.
To determine the margin of error of your data sets, you need to ensure that the selected research sample truly represents your population of interest. This starts from identifying the right population of interest to applying the appropriate sampling technique.
Once you have a well-representative sample, you can go ahead to calculate the margin of error using the following formula:
Where;
P-hat = Sample Proportion
n = Sample Size
z = Z-Score
Here’s an example of how to apply this formula to your data.
Let’s say you conducted a poll on how many people use Snapchat for business. Out of 1,000 respondents, 450 chose “yes.” You also have a confidence level of 90%.
Step 1: Determine P-hat
P-hat = Numbers of respondents who chose yes ÷ Total number of respondents
P-hat = 450 ÷ 1000 = 0.45 or 45%
Step 2: Find the Z-Score of the corresponding confidence level. Using this Z-Score table, the Z-score of 90 is 1.645.
Step 3: Apply the Margin of Error formula.
Margin of Error = Z-score × [√P-hat (1–P-hat)] ÷ n]
1.645 × [√(0.45 × 0.55) ÷ 1000]
Margin of Error = 2.6%
Alternatively, you can calculate the margin of error using the standard deviation of your data. This means:
Margin of Error = Z-Score × (S ÷ √n)
If you’d rather not bother with manual calculations, you can use this margin of error calculator to determine the value for your data set.
What does it mean when your data has a margin of error to a certain percentage? More importantly, how do you interpret this in layman terms to an audience that isn’t “statistics savvy?”
The first thing you should communicate is that margin of error accounts for the maximum difference between outcomes from your sample size and that of the population of interest. So, if you have a margin of error of ±3%, it means the actual results from your target population could be 3% more or less than the estimated value.
For example, suppose your survey shows that 62% of your sample size smoke cigarettes. When you extrapolate this data to the actual population, you would have to add and subtract the margin of error, which means the actual data would fall between 59% and 65%.
Arithmetic mean is a statistical measure of distribution that accounts for the average of all the variables in a data set. In other words, arithmetic mean allows you to find the central value of a finite set of variables.
Often, you’d need the mean of your day sets for calculating the measure of error, especially if you need to determine the standard deviation of your data first.
Arithmetic Mean = Sum of all Variables ÷ Number of Variables
Suppose you have a date set with the following values: 13, 5, 19, 7, 12.
Arithmetic Mean = (13+5+19+7+12) ÷ 5 = 11.2
Confidence interval and confidence level are often used interchangeably, but they do not mean the same thing. Let’s look at some key differences.
A confidence interval is a range of results from a poll, experiment, or survey that the researcher expects to contain the population parameter of interest. Typically, researchers use the confidence interval of observation to determine whether a parameter will fall between a pair of values around the mean.
On the other hand, the confidence level is the degree of certainty or probability that a survey will produce the same results repeatedly. Theoretically, a confidence level is a measure of accuracy for research outcomes and survey results.
Confidence levels are measured in percentages, while confidence intervals are measured in ranges. This means that the confidence level of a data set can be 90% while the confidence interval is 45±50.
Confidence Interval (C.I) = X ± Z × (S÷√n)
Where;
X = Arithmetic Mean
Z = Confidence Level Value
S = Sample Standard Deviation
n = Sample Size
Since you cannot calculate the confidence level of any observation, most researchers choose an estimated value based on the type of research, sampling technique, and the like. Many times, this ends up as 95%.
Confidence interval attaches a measure of accuracy to your sample data. It also helps you to evaluate the reliability of the different variables in an observation.
Like confidence intervals, confidence level allows you to measure the level of accuracy of your data. In addition, it helps you to know if the sample mean is a good or poor representation of the population mean.
The standard deviation of observation is the amount of variation or dispersion of values around its mean. In other words, it tells you how far apart each value or variable lies from the mean of the same data set. Typically, the higher the standard deviation, the more dispersed the data is.
Understanding standard deviation is an essential part of calculating the margin of error for your data. If you do not have the value for your sample data proportion, you can use standard deviation to determine the margin of error.
Where;
s = sample standard deviation
∑ = sum of…
X = each value
x̅ = sample mean
n = number of values in the sample
Also referred to as population of interest, population size is the entire group with the measurable quality or parameter your research is based on. For example, if you’re researching birds of prey in Africa, your population size would be all kinds of birds of prey on the continent.
The sample size is the subset of your population of interest who become direct participants in the research. For example, out of all the birds of prey, you could narrow your research to the African Hawk Eagle in Angola.
A critical value is the split point in hypothesis testing that determines whether the researcher accepts or rejects the null hypothesis. When plotted on a graph, the critical value separates the variables into several sections, including the rejection regions. When calculating the margin of error, researchers use the critical value to determine the applicable ranges within the data sets.
6. Z-score
Z-score or standard score accounts for the difference between a given data point and the mean in an observation. It accounts for the standardization of the different data points in the distribution, allowing you to compare scores on other variables.
Because of their similarities, it’s pretty easy for students and researchers to mistake the margin of error for standard error and vice versa. In this section, we’ll look at some key differences between standard error and margin of error. Let’s start with their definitions.
Definition
A margin of error is a statistical measure that accounts for the degree of error received from the outcome of your research sample. On the other hand, standard error measures the accuracy of the representation of the population sample to the mean using the standard deviation of the data set.
Purpose
The purpose of the standard error is to measure the spread of random variables within your data set, while the goal of the margin of error is to estimate how much allowable difference can exist between the research population and sample size.
Formula
Margin of Error = Z-score × [√P-hat (1–P-hat)] ÷ n]
OR
Margin of Error = Z-Score × (S ÷ √n)
Where;
P-hat = Sample Proportion
n = Sample Size
z = Z-Score
S = Standard Deviation
Standard Error = Sample Standard Deviation ÷ √number of samples
From all we’ve discussed so far, you should already have a fair idea of the factors that primarily affect the margin of error. These factors are confidence level, sample size, and standard deviation.
It follows that increasing or decreasing any of them will have a dominant effect on the value of margin of error for your data. For example, if your sample size goes from 1,000 to 10,000, your margin of error might increase or reduce to a similar magnitude.
Since a lower margin means a higher level of accuracy in your research results and sample, it follows that many researchers strive to reduce the margin of error in their systematic investigation. But how do you do this? Here are some ideas you can test out.
Whether in polling, market research, or simple data collection, a margin of error works in the same way—helping you determine the accuracy of your sample in relation to the actual population of interest. Let’s consider specific applications.
In polling, a margin of error allows you to accurately extrapolate results for one option to a broader population, with minimal inaccuracy. Here’s what we mean.
Let’s say you conduct a popularity poll for two candidates. Based on the data, you peg the margin of error for Candidate A at 3%. It means there’s a close approximation between the results and the actual popularity of the candidate in the population of interest. So, you can accurately predict future variables using present polling results.
In market research, the margin of error shows the level of confidence an organization should have in the data collected from surveys. As mentioned earlier, the higher the margin of error, the less confident you should be in the research samples and, ultimately, the results obtained.
Conclusion
In this article, we’ve discussed the margin of error and its influence on the outcomes of any systematic investigation. Since research is based on collecting samples from a population of interest, you must account for variations between your sample data and the target audience. Calculating the margin of error of your observation is the most effective way to achieve this.
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