At 95 confidence what is the margin of error

Introduction

While you are learning statistics, you will often have to focus on a sample rather than the entire population. This is because it is extremely costly, difficult and time-consuming to study the entire population. The best you can do is to take a random sample from the population – a sample that is a ‘true’ representative of it. You then carry out some analysis using the sample and make inferences about the population.

Since the inferences are made about the population by studying the sample taken, the results cannot be entirely accurate. The degree of accuracy depends on the sample taken – how the sample was selected, what the sample size is, and other concerns. Common sense would say that if you increase the sample size, the chances of error will be less because you are taking a greater proportion of the population. A larger sample is likely to be a closer representative of the population than a smaller one.

Let’s consider an example. Suppose you want to study the scores obtained in an examination by students in your college. It may be time-consuming for you to study the entire population, i.e. all students in your college. Hence, you take out a sample of, say, 100 students and find out the average scores of those 100 students. This is the sample mean. Now, when you use this sample mean to infer about the population mean, you won’t be able to get the exact population means. There will be some “margin of error”.

You will now learn the answers to some important questions: What is margin of error, what are the method of calculating margins of error, how do you find the critical value, and how to decide on t-score vs z-scores. Thereafter, you’ll be given some margin of error practice problems to make the concepts clearer.

What is Margin of Error?

The margin of error can best be described as the range of values on both sides (above and below) the sample statistic. For example, if the sample average scores of students are 80 and you make a statement that the average scores of students are 80 ± 5, then here 5 is the margin of error.

Calculating Margins of Error

For calculating margins of error, you need to know the critical value and sample standard error. This is because it’s calculated using those two pieces of information.

The formula goes like this:

margin of error = critical value * sample standard error.

How do you find the critical value, and how to calculate the sample standard error? Below, we’ll discuss how to get these two important values.

How do You find the Critical Value?

For finding critical value, you need to know the distribution and the confidence level. For example, suppose you are looking at the sampling distribution of the means. Here are some guidelines.

1. If the population standard deviation is known, use z distribution.
2. If the population standard deviation is not known, use t distribution where degrees of freedom = n-1 (n is the sample size). Note that for other sampling distributions, degrees of freedom can be different and should be calculated differently using appropriate formula.
3. If the sample size is large, then use z distribution (following the logic of Central Limit Theorem).

It is important to know the distribution to decide what to use – t-scores vs z-scores.

Caution – when your sample size is large and it is not given that the distribution is normal, then by Central Limit Theorem, you can say that the distribution is normal and use z-score. However, when the sample size is small and it is not given that the distribution is normal, then you cannot conclude anything about the normality of the distribution and neither z-score nor t-score can be used.

When finding the critical value, confidence level will be given to you. If you are creating a 90% confidence interval, then confidence level is 90%, for 95% confidence interval, the confidence level is 95%, and so on.

Here are the steps for finding critical value:

Step 1: First, find alpha (the level of significance). alpha =1 – Confidence level.

For 95% confidence level, alpha =0.05

For 99% confidence level, alpha =0.01

Step 2: Find the critical probability p*. Critical probability will depend on whether we are creating a one-sided confidence interval or a two-sided confidence interval.

For two-sided confidence interval, p*=1-dfrac { alpha }{ 2 }

For one-sided confidence interval, p*=1-alpha

Then you need to decide on using t-scores vs z-scores. Find a z-score having a cumulative probability of p*. For a t-statistic, find a t-score having a cumulative probability of p* and the calculated degrees of freedom. This will be the critical value. To find these critical values, you should use a calculator or respective statistical tables.

Sample Standard Error

Sample standard error can be calculated using population standard deviation or sample standard deviation (if population standard deviation is not known). For sampling distribution of means:

Let sample standard deviation be denoted by s, population standard deviation is denoted by sigma and sample size be denoted by n.

text {Sample standard error}=dfrac { sigma }{ sqrt { n } }, if sigma is known

text {Sample standard error}=dfrac { s }{ sqrt { n } }, if sigma is not known

Depending on the sampling distributions, the sample standard error can be different.

Having looked at everything that is required to create the margin of error, you can now directly calculate a margin of error using the formula we showed you earlier:

Margin of error = critical value * sample standard error.

Some Relationships

1. Confidence level and marginal of error

As the confidence level increases, the critical value increases and hence the margin of error increases. This is intuitive; the price paid for higher confidence level is that the margin of errors increases. If this was not so, and if higher confidence level meant lower margin of errors, nobody would choose a lower confidence level. There are always trade-offs!

2. Sample standard deviation and margin of error

Sample standard deviation talks about the variability in the sample. The more variability in the sample, the higher the chances of error, the greater the sample standard error and margin of error.

3. Sample size and margin of error

This was discussed in the Introduction section. It is intuitive that a greater sample size will be a closer representative of the population than a smaller sample size. Hence, the larger the sample size, the smaller the sample standard error and therefore the smaller the margin of error.

Margin of Error Practice Problems

Example 1

25 students in their final year were selected at random from a high school for a survey. Among the survey participants, it was found that the average GPA (Grade Point Average) was 2.9 and the standard deviation of GPA was 0.5. What is the margin of error, assuming 95% confidence level? Give correct interpretation.

Step 1: Identify the sample statistic.

Since you need to find the confidence interval for the population mean, the sample statistic is the sample mean which is the average GPA = 2.9.

Step 2: Identify the distribution – t, z, etc. – and find the critical value based on whether you need a one-sided confidence interval or a two-sided confidence interval.

Since population standard deviation is not known and the sample size is small, use a t distribution.

text {Degrees of freedom}=n-1=25-1=24.

alpha=1-text {Confidence level}=1-0.95=0.05

Let the critical probability be p*.

For two-sided confidence interval,

p*=1-dfrac { alpha }{ 2 } =1-dfrac { 0.05 }{ 2 } =0.975.

The critical t value for cumulative probability of 0.975 and 24 degrees of freedom is 2.064.

Step 3: Find the sample standard error.

text{Sample standard error}=dfrac { s }{ sqrt { n } } =dfrac { 0.5 }{ sqrt { 25 } } =0.1

Step 4: Find margin of error using the formula:

Margin of error = critical value * sample standard error

= 2.064 * 0.1 = 0.2064

Interpretation: For a 95% confidence level, the average GPA is going to be 0.2064 points above and below the sample average GPA of 2.9.

Example 2

400 students in Princeton University are randomly selected for a survey which is aimed at finding out the average time students spend in the library in a day. Among the survey participants, it was found that the average time spent in the university library was 45 minutes and the standard deviation was 10 minutes. Assuming 99% confidence level, find the margin of error and give the correct interpretation of it.

Step 1: Identify the sample statistic.

Since you need to find the confidence interval for the population mean, the sample statistic is the sample mean which is the mean time spent in the university library = 45 minutes.

Step 2: Identify the distribution – t, z, etc. and find the critical value based on whether the need is a one-sided confidence interval or a two-sided confidence interval.

The population standard deviation is not known, but the sample size is large. Therefore, use a z (standard normal) distribution.

alpha=1-text{Confidence level}=1-0.99=0.01

Let the critical probability be p*.

For two-sided confidence interval,

p*=1-dfrac { alpha }{ 2 } =1-dfrac { 0.01 }{ 2 } =0.995.

The critical z value for cumulative probability of 0.995 (as found from the z tables) is 2.576.

Step 3: Find the sample standard error.

text{Sample standard error}=dfrac { s }{ sqrt { n } } =dfrac { 10 }{ sqrt { 400 } } =0.5

Step 4: Find margin of error using the formula:

Margin of error = critical value * sample standard error

= 2.576 * 0.5 = 1.288

Interpretation: For a 99% confidence level, the mean time spent in the library is going to be 1.288 minutes above and below the sample mean time spent in the library of 45 minutes.

Example 3

Consider a similar set up in Example 1 with slight changes. You randomly select X students in their final year from a high school for a survey. Among the survey participants, it was found that the average GPA (Grade Point Average) was 3.1 and the standard deviation of GPA was 0.7. What should be the value of X (in other words, how many students you should select for the survey) if you want the margin of error to be at most 0.1? Assume 95% confidence level and normal distribution.

Step 1: Find the critical value.

alpha=1-text{Confidence level}=1-0.95=0.05

Let the critical probability be p*.

For two-sided confidence interval,

p*=1-dfrac { alpha }{ 2 } =1-dfrac { 0.05 }{ 2 } =0.975.

The critical z value for cumulative probability of 0.975 is 1.96.

Step 3: Find the sample standard error in terms of X.

text{Sample standard error}=dfrac { s }{ sqrt { X } }=dfrac { 0.7 }{ sqrt { X } }

Step 4: Find X using margin of error formula:

Margin of error = critical value * sample standard error

0.1=1.96*dfrac { 0.7 }{ sqrt { X } }

This gives X=188.24.

Thus, a sample of 189 students should be taken so that the margin of error is at most 0.1.

Conclusion

The margin of error is an extremely important concept in statistics. This is because it is difficult to study the entire population and the sampling is not free from sampling errors. The margin of error is used to create confidence intervals, and most of the time the results are reported in the form of a confidence interval for a population parameter rather than just a single value. In this article, you made a beginning by learning answering questions like what is margin of error, what is the method of calculating margins of errors, and how to interpret these calculations. You also learned to decide whether to use t-scores vs z-scores and gained information about finding critical values. Now you know how to use margin of error for constructing confidence intervals, which are widely used in statistics and econometrics.

Let’s put everything into practice. Try this Statistics practice question:

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If you want to solve some confidence interval problems, you’re in the right place. Our 95% confidence interval calculator will help you calculate this confidence interval and provide you with the essential knowledge! Read on to learn:

• What is the 95% confidence interval formula;
• What is the interpretation of the 95% confidence interval (or any chosen one, to be honest); and
• What is the p-value for the 95 percent confidence interval?

95% confidence interval — formula and interpretation

To calculate a 95% confidence interval, we first need to calculate the standard error. We can use the formula:

SE=σnSE=frac{σ}{sqrt{n}}

Now let’s estimate the margin of error.

ME=SE×Z(0.95)ME = SE × Z(0.95)

upper bound=μ+MEtext{upper} text{bound} = μ + ME

lower bound=μ−MEtext{lower} text{bound} = μ — ME

Interpretation of the 95% confidence interval

How should you understand the results? Let’s imagine George wants to calculate the mean height of his family members. Some relatives live very far, and George cannot measure them all. But he managed to get 30 measurements, and the average height of those measurements was 172 cm (5 ft 7 in). He found out, that the 95% confidence interval range is 161-183 cm (5 ft 3 in — 6 ft). Based on those measurements, he can be 95% sure that any family member’s height falls between 162-183 cm.

Using 95% confidence interval calculator

Confidence interval calculators

FAQ

How do I calculate a 95% confidence interval?

What is the p-value at 95 percent confidence interval?

The uncorrected p-value at 95% confidence interval is 0.05.

From Wikipedia, the free encyclopedia

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result would reflect the result of a census of the entire population. The margin of error will be positive whenever a population is incompletely sampled and the outcome measure has positive variance, which is to say, the measure varies.

The term margin of error is often used in non-survey contexts to indicate observational error in reporting measured quantities.

Concept[edit]

Consider a simple yes/no poll as a sample of respondents drawn from a population reporting the percentage of yes responses. We would like to know how close is to the true result of a survey of the entire population , without having to conduct one. If, hypothetically, we were to conduct poll over subsequent samples of respondents (newly drawn from ), we would expect those subsequent results to be normally distributed about . The margin of error describes the distance within which a specified percentage of these results is expected to vary from .

According to the 68-95-99.7 rule, we would expect that 95% of the results will fall within about two standard deviations () either side of the true mean .  This interval is called the confidence interval, and the radius (half the interval) is called the margin of error, corresponding to a 95% confidence level.

Generally, at a confidence level , a sample sized of a population having expected standard deviation has a margin of error

where denotes the quantile (also, commonly, a z-score), and is the standard error.

Standard deviation and standard error[edit]

We would expect the normally distributed values   to have a standard deviation which somehow varies with . The smaller , the wider the margin. This is called the standard error .

For the single result from our survey, we assume that , and that all subsequent results together would have a variance .

Note that corresponds to the variance of a Bernoulli distribution.

Maximum margin of error at different confidence levels[edit]

For a confidence level , there is a corresponding confidence interval about the mean , that is, the interval within which values of should fall with probability . Precise values of are given by the quantile function of the normal distribution (which the 68-95-99.7 rule approximates).

Note that is undefined for , that is, is undefined, as is .

0.68	0.994457883210	0.999	3.290526731492
0.90	1.644853626951	0.9999	3.890591886413
0.95	1.959963984540	0.99999	4.417173413469
0.98	2.326347874041	0.999999	4.891638475699
0.99	2.575829303549	0.9999999	5.326723886384
0.995	2.807033768344	0.99999999	5.730728868236
0.997	2.967737925342	0.999999999	6.109410204869

Since at , we can arbitrarily set , calculate , , and to obtain the maximum margin of error for at a given confidence level and sample size , even before having actual results.  With

Also, usefully, for any reported

Specific margins of error[edit]

If a poll has multiple percentage results (for example, a poll measuring a single multiple-choice preference), the result closest to 50% will have the highest margin of error. Typically, it is this number that is reported as the margin of error for the entire poll. Imagine poll reports as

(as in the figure above)

As a given percentage approaches the extremes of 0% or 100%, its margin of error approaches ±0%.

Comparing percentages[edit]

Imagine multiple-choice poll reports as . As described above, the margin of error reported for the poll would typically be , as is closest to 50%. The popular notion of statistical tie or statistical dead heat, however, concerns itself not with the accuracy of the individual results, but with that of the ranking of the results. Which is in first?

If, hypothetically, we were to conduct poll over subsequent samples of respondents (newly drawn from ), and report result , we could use the standard error of difference to understand how is expected to fall about . For this, we need to apply the sum of variances to obtain a new variance, ,

where is the covariance of and .

Thus (after simplifying),

Note that this assumes that is close to constant, that is, respondents choosing either A or B would almost never chose C (making and close to perfectly negatively correlated). With three or more choices in closer contention, choosing a correct formula for becomes more complicated.

Effect of finite population size[edit]

The formulae above for the margin of error assume that there is an infinitely large population and thus do not depend on the size of population , but only on the sample size . According to sampling theory, this assumption is reasonable when the sampling fraction is small. The margin of error for a particular sampling method is essentially the same regardless of whether the population of interest is the size of a school, city, state, or country, as long as the sampling fraction is small.

In cases where the sampling fraction is larger (in practice, greater than 5%), analysts might adjust the margin of error using a finite population correction to account for the added precision gained by sampling a much larger percentage of the population. FPC can be calculated using the formula^[1]

…and so, if poll were conducted over 24% of, say, an electorate of 300,000 voters,

Intuitively, for appropriately large ,

In the former case, is so small as to require no correction. In the latter case, the poll effectively becomes a census and sampling error becomes moot.

See also[edit]

• Engineering tolerance
• Key relevance
• Measurement uncertainty
• Random error

References[edit]

Sources[edit]

• Sudman, Seymour and Bradburn, Norman (1982). Asking Questions: A Practical Guide to Questionnaire Design. San Francisco: Jossey Bass. ISBN 0-87589-546-8
• Wonnacott, T.H.; R.J. Wonnacott (1990). Introductory Statistics (5th ed.). Wiley. ISBN 0-471-61518-8.

External links[edit]

• «Errors, theory of», Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. «Margin of Error». MathWorld.

In a

confidence interval, the range of values above and below the
sample statistic is called the margin of error.

For example, suppose we wanted to know the percentage of adults that
exercise daily. We could devise a

sample design to ensure that our sample estimate will not
differ from the true population value by more than,
say, 5 percent 90 percent of the time
(the

confidence level). In this example, the margin of error would be 5 percent.

How to Compute the Margin of Error

The margin of error can be defined by either of the following
equations.

If you know the standard deviation of the statistic, use the first
equation to compute the margin of error. Otherwise, use the second
equation. In a previous lesson, we described

how to compute the standard deviation and standard error.

How to Find the Critical Value

The critical value is a factor used to compute
the margin of error. This section describes how to find the
critical value, when the

sampling distribution
of the statistic is
normal
or nearly normal.

When the sampling distribution is nearly normal, the critical value can be expressed as a
t score or as a
z-score.
To find the critical value, follow these steps.

T-Score vs. Z-Score

Should you express the critical value as a t statistic or as a z-score? One way to answer
this question focuses on the population standard deviation.

• If the population standard deviation is unknown, use the t statistic.

Another approach focuses on sample size.

• If the sample size is small, use the t statistic.

In practice, researchers employ a mix of the above guidelines. On this site, we use z-scores
when the population standard deviation is known and the sample size is large.
Otherwise, we use t statistics, unless the sample size is small and the underlying
distribution is not normal.

Warning: If the sample size is small and the population distribution is not normal,
we cannot be confident that the sampling distribution of the statistic will be normal. In this
situation, neither the t statistic nor the z-score should be used to compute critical values.

You can use the
Normal Distribution Calculator
to find the critical z-score, and the
t Distribution Calculator to find
the critical t statistic. You can also use a graphing calculator or
standard statistical tables (found in the appendix of
most introductory statistics texts).

Test Your Understanding

Problem 1

Nine hundred (900) high school freshmen were randomly selected for
a national survey. Among survey participants, the mean grade-point
average (GPA) was 2.7, and the standard deviation was 0.4. What
is the margin of error, assuming a 95% confidence level?

(A) 0.013

(B) 0.025

(C) 0.500

(D) 1.960

(E) None of the above.

Solution

The correct answer is (B). To compute the margin of error, we
need to find the
critical value and the standard error of the mean.
To find the critical value, we take the following steps.

• Find the critical value.
  Since we don’t know the population standard deviation, we’ll express the
  critical value as a t statistic. For this problem, it will
  be the t statistic having 899 degrees of freedom and
  a cumulative probability equal to 0.975. Using the
  t Distribution Calculator,
  we find that the critical value is about 1.96.

Next, we find the standard error of the mean, using the following
equation:

SE_x = s / sqrt( n )

SE_x = 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

And finally, we compute the margin of error (ME).

ME = Critical value x Standard error

ME = 1.96 * 0.013 = 0.025

This means we can be 95% confident that the mean grade point average
in the population is 2.7 plus or minus 0.025, since the margin of error
is 0.025.

Note: The larger the sample size, the more closely the t distribution
looks like the normal distribution. For this problem, since the sample size is very large, we would have
found the same result with a z-score as we found with a t statistic. That is,
the critical value would still have been about 1.96. The choice of t statistic versus z-score does not
make much practical difference when the sample size is very large.

The margin of error is an important measure in statistics. The degree of error in random sampling surveys is known as the margin of error. It expresses random sampling error in the results of a survey. It depicts the confidence that the researchers should have in the data obtained from the survey. It tells the variance from the actual population in terms of the percentage points. A 95% confidence interval with a 4% margin of error, for instance, indicates that your statistic will, 95% of the time, be within 4% of the true population figure. A wider margin of error in statistics denotes a reduced chance of relying on a survey’s or poll’s findings, meaning that there will be less trust in the results’ ability to accurately reflect a community.

What is a Margin of Error?

A margin of error tells you how many percentages points our results differ from the real value. The margin of error is defined as the range of values below and above the sample statistic in a confidence interval. The confidence interval is a way to show what the uncertainty is with a certain statistic. 

Importance of Margin of Error

The importance of Magin of Error is discussed below in this article:

• The margin of error indicates how close the results would be if the whole population was surveyed instead of a sample population.
• The higher the margin of error, the lesser the chances of relying upon the results of the survey.
• If the margin of error is low, it means that the results obtained from the sample are highly reliable and will be very close to the ones obtained when surveying the complete population.
• The margin of error is a very important measure from the market viewpoint.
• It is a crucial tool for market research since it shows the amount of assurance that should be placed in survey data by the researchers.
• It is used to show how confident a statistician is in determining whether or not the outcomes of an online survey or online poll are representative of the full population.

Calculation of Margin of Error

Each person in a population has a chance of being chosen to be a part of the sample when using probability sampling. By using this technique, statisticians and researchers may choose participants who have expertise in their particular field of study, ensuring the lowest possible margin of error for the data drawn from these samples. Non-probability sampling excludes some segments of the population due to the selection technique used to create the samples, which is based on convenience or cost-effectiveness rather than applicability. Only by separating respondents into groups based on their interests and relevance to the poll at hand will surveys be effective.

In terms of sample size, the formula is given by, 

M.O.E. = 1 / √(n)

where,
M.O.E. denotes the margin of error
n is the sample size

The formula is not widely used as it does not give accurate results it only takes the sample size into consideration and not any other parameters. But in terms of z-index and sample size, the formula is given by,

M.O.E.= z × σ / √(n)

where,
z denotes the z-index
n is the sample size
σ is the population standard deviation

Thus, the margin of error is the product of the critical value and the standard deviation. In the above formula, the z score is calculated using the confidence level given in the question. Some widely used z scores and confidence levels are shown below:

How to Calculate Margin of Error?

Follow the steps discussed below to calculate the Magin of Error.

Step 1: Find the critical value i.e. either a t-score or a z-score. In general, for small sample sizes (under 30) use a t-score otherwise, use a z-score.

Step 2: Find the Standard Deviation or the Standard Error.

Step 3: Multiply the critical value from Step 1 by the standard deviation or standard error from Step 2. 

Example: If your critical value is 1.645 and your standard error is 0.26, then:

Margin of Error = 1.645 × 0.26 
                         = 0.4277

Solved Examples on Margin of Error

Example 1: A random sample of 50 students has average pocket money of 2500 and a standard deviation of 567. Find the margin of error if c = 0.95.

Solution:

Given, n = 50

σ = 567

For c = 0.95 i.e. 95% confidence level, z = 1.96

M.O.E = 1.96 × 567 / √(50)

M.O.E. = 157.16

Example 2: 600 students were surveyed, and it was found that the average CGPA of the students was 8.7, with a population standard deviation of 0.2. Calculate the margin of error for the 90% confidence level.

Solution:

Given, n = 600

σ = 0.2

For confidence level of 90%, z = 1.645

M.O.E = 1.645 × 0.2 / √(600)

M.O.E. = 0.013

Example 3: In a survey, it was found that the average salary of 5000 employees was 350000, with a population standard deviation of 0.3. What is the margin of error at a 99% confidence level?

Solution:

Given, n = 5000

σ = 0.3

For 99% confidence level, z = 2.575

M.O.E = 2.575 × 0.3 / √(5000)

M.O.E. = 0.0109

Example 4: Find the margin of error when a population of 400 people is surveyed.

Solution:

Given, sample size = 400

∴M.O.E. = 1 / √(400)

               = 1/20

               = 0.05

FAQs on Margin of Error

Question 1: What do you mean by the margin of error?

Solution:

The margin of error shows how similar the findings would be if the entire population had been polled as opposed to just a sample.

Question 2: How is the z-value determined in statistics?

Solution:

Z-score is a metric that quantifies how closely a value relates to the mean of a set of values. Standard deviations of data from their mean are used to calculate Z-score.

Question 3: Can the margin of error have a negative value?

Solution:

The level of accuracy needed in a survey is called the margin of error. For example, a 8% margin of error indicates that the actual results’ variation might be either negative or positive eight points.

Question 4: How can we figure out the margin of error with an 85% level of confidence?

Solution:

It is known that the error value is 0.0219 or z-value, is 1.96 for a 95% confidence level. As a result, the method above may be used to calculate the margin of error at a 85% confidence level.

Related Resources

• Statistics Formulas
• Variance and Standard Deviation
• Frequency Distributions