Confidence Interval Formula

What is the Confidence Interval Formula?

The confidence interval assesses the level of uncertainty with specific statistics and is employed along with the margin of error. The confidence interval selection for a given interval computes the probability that the resulting confidence interval would contain the true parameter value.

Confidence intervals are inherently related to confidence levels.  The confidence interval is determined using normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more, T-distribution, and employing proportions. A true population parameter is defined as the value that represents the characteristic of the specific population. The confidence interval equation in general form would be represented as follows: –

Confidence Interval Formula = Mean of Sample ± Critical Factor × Standard Deviation of Sample
Confidence-Interval-Formula

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Explanation of the Confidence Interval Formula

The confidence interval equation can be calculated by using the following steps:

Step 1: Firstly, determine the criteria or phenomenon to be taken up for testing. It would be seen how close the predictions would lie with respect to the chosen criterion.

Step 2: Next, from the population, shortlist, or choose the sample from it. The data gathered or sample formulated would be utilized for the purpose of testing or performing the hypothesis.

Step 3: Next, for the sample chosen, determine mean and standard deviationStandard DeviationStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability.read more. This would help determine the population parameter.

Step 4: Next, Determine the confidence level. The confidence level may range from 90 percent to 99 percent. For example, If the confidence level is chosen for 95 percent, then it is inferred that the analyst is sure for 95 percent that the parameter is contained in the sample chosen.

Step 5: Now, Determine the confidence coefficient for the confidence interval chosen for the determination of confidence interval. To determine the confidence coefficient, for the value of confidence level, refer to the corresponding table for the coefficient. Suppose the confidence coefficient is determined using z-tables wherein the analyst can refer the table to arrive at the critical value or the coefficient.

Step 6: Now, Determine the margin of errorMargin Of ErrorThe margin of error is a statistical expression to determine the percentage point the result arrived at will differ from the actual value. Standard deviation divided by the sample size, multiplying the resultant figure with the critical factor. Margin of Error = Z * ơ / √nread more. The margin of error is expressed as displayed below: –

The margin of Error = Critical Factor × Standard deviation of the sample.

  • Margin of Error = Za/2 × σ/√(n)

Here,

  • The critical value of the sample is represented as Za/2.
  • The sample size is represented as n.
  • The standard deviation is represented as σ.

Step 7: Now, Determine the Confidence interval for the chosen sample with the confidence level. The confidence interval formula is expressed as displayed below: –

Confidence Interval=Mean of Sample ± Critical Factor × Standard deviation of the Sample.

Examples of Confidence Interval Formula

Let’s see some simple to advanced practical examples of the confidence interval equation to understand it better.

You can download this Confidence Interval Formula Excel Template here – Confidence Interval Formula Excel Template

Confidence Interval Formula – Example #1

Let us take the example of a university that is assessing the average height of students on board with the university. The management determined the average height of the students undertaken in the batch is 170 cm. The batch strength is 1,000 students, and the standard deviation among the students is broadly at 20 cm. 

Help the university management determine the confidence interval on the average height of the students on board with the university. Assume confidence level to be at 95 percent.

Use below given data for the calculation of confidence interval.

Confidence Interval Formula Example 1

Calculation of Margin of Error using below formula is as follows,

Confidence Interval Formula Example 1.1
  • Margin of Error = Za/2 × σ/√(n)
  • = 1.96 × 20/√ (1,000)
  • = 1.96 × 20/31.62
  • = 1.96 × 0.632
  • Margin of Error = 1.2396

Calculation of Confidence Interval at level 1

Confidence Interval Formula Example 1.2

Confidence Interval=Mean of Sample ± Margin of Error

=170 ± 1.2396

Confidence value = 170 + 1.2396

Confidence Interval at level 1 will be  –

Confidence Interval Formula Example 1.3
  • Confidence Interval value at level 1 = 171.2396

Calculation of Confidence Interval at level 2

Example 1.4

=Confidence value = 170 – 1.2396

Confidence Interval at level 2 will be  –

Example 1.5
  • Confidence Interval Value at level 2 = 168.7604

Therefore, both the confidence interval for the average height of students is 168.7604 cm to 171.2396 cm.

Confidence Interval Formula – Example #2

Let us take the example of a hospital that is trying to assess the confidence interval on the number of patients received by it during the month. The management determined the average number of patients received for the month is 2,000 people. The hospital has a capacity of 4,000 patients, and the standard deviation among the students is broadly at 1000 individuals. 

Help the university management determine the confidence interval on the average height of the students on board with the university. Assume confidence level to be at 95 percent.

Use below given data for the calculation of confidence interval.

Example 2

Calculation of Margin of Error using below formula is as follows,

Confidence Interval Formula Example 2.1
  • Margin of Error = Za/2 × σ/√(n)
  • = 1.96 × 1,000/√ (4,000)
  • = 1.96 × 1,000/ 63.25
  • = 1.96 × 15.811
  • Margin of Error = 30.99

Calculation of Confidence Interval at level 1

Confidence Interval Formula Example 2.2

Confidence Interval=Mean of Sample ± Margin of Error

  • Confidence Interval=2,000 ± 30.99
  • Confidence Value = 2,000 + 30.99

Confidence Interval at level 1 will be –

Confidence Interval Formula Example 2.3
  • Confidence Interval Value at level 1 = 2031.0

Calculation of Confidence Interval at level 2

Confidence Interval Formula Example 2.4
  • = Confidence value = 2000 – 30.99

Confidence Interval at level 2 will be –

Confidence Interval Formula Example 2.5
  • Confidence Interva Value at level 2 = 1969.0

Therefore, both the confidence interval for the average patients received by the hospital is 1969 individuals to 2,031 individuals.

Relevance and Uses

The application of the confidence interval is to provide a series of values for the population undertaken instead of the estimate of pointEstimate Of PointA point estimator is a statistical function used to derive an approximate single value which serves as a base to estimate the unknown population parameter among the sample data set of the whole population. It is considered to be an unbiased, consistent and most efficient technique.read more or a single value. It further helps in determining that the confidence interval may not contain the value or estimate being look upon, but the probability of finding that specific estimate would be more than the probability of not finding that specific estimate from the range of values chosen in the confidence interval.

For each confidence interval, it is necessary to choose the confidence level for determining whether the estimate lies in the confidence level. A confidence level undertaken could be 90%, 95%, or 99%. For most of the analysis, a confidence level of 95 percent is undertaken that is further used to determine the confidence coefficient and thereby the confidence interval.

Confidence Interval Formula in Excel (with Excel Template)

Now, let us take the excel example to illustrate the concept of the confidence interval in the excel template below. Let us consider the example 1 in the excel to further illustrate the concept of a confidence interval formula. The table provides the detailed explanation of the confidence interval-

Similarly, A cricket team is trying to determine the confidence level of the average weight of the players in the squad. The squad has a sample size of 15 members. Assume the confidence level to be at 95 percent. For a confidence level of 95 percent, the confidence coefficient is determined at 1.96. The sample sizeSample SizeThe sample size formula depicts the relevant population range on which an experiment or survey is conducted. It is measured using the population size, the critical value of normal distribution at the required confidence level, sample proportion and margin of error.read more for the analysis is displayed below.

Confidence Interval Formula Example 3

The first step involves the determination of the average weight of the sample as displayed below: –

Example 3.1

The following would be the results on the above calculation: –

Average

Example 3.2
  • Average = 73.067

The second step involves the determination of the standard deviation on the weight of the sample as displayed below: –

STDEV

Example 3.3

The following would be the results on the above calculations: –

Example 3.4
  • STDEV (Standard Deviation) = 13.2

The third step involves the determination of the margin on an error on the weight of the sample as displayed below: –

Margin of Error

Confidence Interval Formula Example 3.5

The following would be the results on the above calculations: –

Confidence Interval Formula Example 3.6
  • Margin of Error = 6.70

Finally, determine the confidence interval as displayed below: –

Calculation of Confidence Interval at level 1

Example 3.7

Confidence Interval = Mean of Sample ± Margin of Error

Confidence Interval=73.067 ± 6.70

  • =73.067+6.70
  • =79.763

Calculation of Confidence Interval at level 2 –

Example 3.8
  • =73.067-6.70
  • =66.371

Therefore, both the confidence interval for the average weight of cricketers in the squad as determined by the management is 79.763 individuals to 66.371 individuals.

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This has been a guide to the Confidence Interval Formula. Here we provide step by step calculation of confidence interval in excel along with practical examples and a downloadable excel template. You can learn more about accounting and budgeting from the following articles –

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