Confidence Interval Calculator

A Confidence Interval Calculator is a free online statistics tool that helps estimate a range of plausible values for a population parameter based on sample data.

Instead of reporting only a sample mean, a confidence interval provides a range around that estimate. The range reflects uncertainty associated with estimating a population value from a sample.

Calculator Pool’s free Confidence Interval Calculator online uses a sample mean, standard deviation, sample size, and selected confidence level to calculate:

• Confidence Interval
• Standard Error
• Margin of Error
• Lower Confidence Limit
• Upper Confidence Limit
• Z Critical Value

For example, suppose:

Sample Mean = 50

Standard Deviation = 10

Sample Size = 25

and:

Confidence Level = 95%

The calculator gives an approximate confidence interval of:

46.0801 to 53.9199

This result is based on a normal critical-value approximation.

Important: A confidence interval is an interval-estimation tool, not a guarantee that a particular population value falls inside the interval. The interpretation depends on the sampling procedure, statistical assumptions, and confidence-interval method being used.

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What Is a Confidence Interval?

A confidence interval (CI) is a range calculated from sample data that is used to estimate an unknown population parameter.

For a population mean, a confidence interval is generally written as:

Estimate ± Margin of Error

For a normal-approximation interval:

CI = Mean ± z × SE*

where:

• Mean = sample mean
• z* = critical z-value
• SE = standard error

The lower and upper values form the confidence interval.

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What Is a Confidence Interval Calculator?

A confidence interval calculator is an online statistics calculator that automatically calculates an estimated interval from sample information.

Calculator Pool’s tool requires:

Sample Mean

Standard Deviation

Sample Size

and:

Confidence Level

It then calculates the corresponding:

Standard Error

Z Critical Value

Margin of Error

Lower Limit

Upper Limit

This makes the calculator useful for statistics, research, data analysis, education, and basic inferential-statistics calculations.

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How to Use the Confidence Interval Calculator

Using Calculator Pool’s online confidence interval calculator is simple.

Step 1: Enter the Sample Mean

Enter the average calculated from your sample.

For example:

50

Step 2: Enter Standard Deviation

Enter the standard deviation of the sample.

For example:

10

Step 3: Enter Sample Size

Enter the number of observations.

For example:

25

Step 4: Select Confidence Level

Calculator Pool supports:

• 80%
• 90%
• 95%
• 98%
• 99%
• 99.5%
• 99.9%

The most commonly used confidence level in many introductory applications is:

95%

Step 5: Calculate

Click:

Calculate Confidence Interval

The calculator displays the estimated interval and supporting calculations.

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Confidence Interval Formula

For the normal-approximation method used by Calculator Pool:

Confidence Interval = Mean ± z × SE*

The standard error of the mean is:

SE = SD ÷ √n

Therefore:

CI = Mean ± z × (SD ÷ √n)*

Where:

• Mean = sample mean
• SD = standard deviation
• n = sample size
• z* = critical z-value

The calculation produces two boundaries:

Lower Limit = Mean − Margin of Error

Upper Limit = Mean + Margin of Error

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What Is the Margin of Error?

The margin of error represents the distance from the sample estimate to either boundary of a symmetric confidence interval.

For a normal-approximation interval:

Margin of Error = z × SE*

For example, if:

SE = 2

and:

z = 1.96*

then:

Margin of Error = 1.96 × 2

= 3.92

The confidence interval around a mean of 50 would therefore be approximately:

50 − 3.92 = 46.08

and:

50 + 3.92 = 53.92

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What Is Standard Error?

Standard Error (SE) measures the sampling variability of an estimator such as the sample mean.

For the sample mean, it is commonly calculated as:

SE = SD ÷ √n

This means that standard error depends on:

Standard Deviation

and:

Sample Size

Calculator Pool also provides a dedicated Standard Error Calculator.

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Confidence Interval Example

Suppose:

Sample Mean = 50

SD = 10

n = 25

Confidence Level = 95%

Step 1: Calculate Standard Error

SE = 10 ÷ √25

SE = 10 ÷ 5

SE = 2

Step 2: Use the 95% Critical Value

For the normal approximation:

z ≈ 1.96*

Step 3: Calculate Margin of Error

MOE = 1.96 × 2

= 3.92

Step 4: Calculate Lower Limit

50 − 3.92 = 46.08

Step 5: Calculate Upper Limit

50 + 3.92 = 53.92

Therefore:

95% Confidence Interval ≈ 46.08 to 53.92

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Confidence Interval Calculation Table

These are the normal critical values used by Calculator Pool’s current confidence interval tool.

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What Does a 95% Confidence Interval Mean?

A common misunderstanding is that a 95% confidence interval means there is a 95% probability that the fixed population parameter is inside the calculated interval.

In the conventional frequentist interpretation, the confidence level describes the long-run performance of the procedure: if the same sampling method and interval construction were repeated many times, approximately 95% of such intervals would contain the true parameter under the relevant assumptions.

For everyday reporting, a 95% confidence interval can therefore be viewed as a range produced by a procedure designed to have 95% long-run coverage.

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What Does a 90% Confidence Interval Mean?

A 90% confidence interval uses a smaller critical value than a 95% interval.

Under the same sample statistics, this generally produces a narrower interval.

For the normal approximation:

z ≈ 1.645*

Therefore:

MOE = 1.645 × SE

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What Does a 99% Confidence Interval Mean?

A 99% confidence interval uses a larger critical value than a 95% interval.

That produces a wider interval when the other inputs remain unchanged.

For the normal approximation:

z ≈ 2.576*

Therefore:

MOE ≈ 2.576 × SE

This demonstrates the trade-off between confidence level and interval width.

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Confidence Level vs Confidence Interval Width

Increasing the confidence level generally makes the interval wider.

For example, using the same sample:

90% CI → narrower

95% CI → wider

99% CI → even wider

This occurs because a higher confidence level requires a larger critical value.

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Confidence Interval and Sample Size

Sample size has an important effect on confidence interval width.

Because:

SE = SD ÷ √n

a larger sample size generally produces a smaller standard error, assuming standard deviation remains the same.

A smaller standard error leads to a smaller margin of error.

Therefore, larger samples can produce narrower confidence intervals.

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Sample Size Example

Suppose:

SD = 10

n = 25

SE = 10 ÷ 5 = 2

n = 100

SE = 10 ÷ 10 = 1

The standard error has been reduced by half.

If the confidence level remains the same, the margin of error also becomes approximately half as large.

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Confidence Interval and Standard Deviation

A larger standard deviation generally produces a larger standard error when sample size remains fixed.

Since:

SE = SD ÷ √n

doubling SD doubles SE.

That in turn doubles the margin of error when the critical value remains unchanged.

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Confidence Interval and Standard Error

The relationship is:

Margin of Error = Critical Value × Standard Error

Therefore, standard error is one of the main building blocks of a confidence interval.

For example:

SE = 2

z = 1.96*

Then:

MOE = 3.92

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Confidence Interval and Mean

The sample mean is the center of a symmetric normal-approximation confidence interval.

The general form is:

Mean ± Margin of Error

For example:

Mean = 50

MOE = 3.92

Therefore:

CI = 46.08 to 53.92

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Confidence Interval and Z-Score

A confidence interval uses a critical z-value when a normal approximation is appropriate.

This is related to the standard normal distribution and z-scores.

For example, the commonly used 95% two-sided interval uses approximately:

z = 1.96*

Calculator Pool also has a Z-Score Calculator for calculating the standardized position of an individual value.

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Confidence Interval vs Z-Score

These two concepts are related but different.

Z-Score

Describes how far a value is from a mean in standard-deviation units.

Confidence Interval

Provides a range around an estimate.

A z-score is often used as part of the process of constructing a confidence interval under normal-based assumptions.

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Confidence Interval vs Standard Error

Standard Error

Measures the sampling variability of an estimator.

Confidence Interval

Combines the estimate, standard error, and a critical value to create a range.

For example:

CI = Mean ± z × SE*

Therefore, standard error is a component of the confidence interval calculation.

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Confidence Interval vs Margin of Error

These are closely related but not the same.

Margin of Error

The distance from the estimate to the confidence boundary.

Confidence Interval

The full range:

Estimate − MOE

to:

Estimate + MOE

For example:

Mean = 50

MOE = 3.92

Confidence interval:

46.08 to 53.92

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Confidence Interval vs Prediction Interval

A confidence interval for a population mean is about uncertainty in an estimated population parameter.

A prediction interval is generally about the range where a future individual observation may fall.

Prediction intervals are therefore usually wider because they account for both estimation uncertainty and individual observation variability.

They should not be treated as interchangeable.

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Confidence Interval vs Standard Deviation

Standard deviation and confidence interval describe different things.

Standard Deviation

Describes variability among observations.

Confidence Interval

Describes uncertainty around an estimated population parameter using sample data and a selected method.

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Confidence Interval vs Variance

Variance measures squared variability in data.

A confidence interval is an interval estimate for a parameter.

Variance may contribute indirectly through standard deviation and standard error, but it is not itself a confidence interval.

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Confidence Interval for a Mean

A confidence interval for a population mean is commonly calculated using:

Sample Mean ± Critical Value × Standard Error

When the population standard deviation is known or a suitable normal approximation is being used, a z-based interval may be appropriate.

When the population standard deviation is unknown and the sample is small, a t-based method is often more appropriate.

Calculator Pool’s current tool uses the normal critical-value approach described above.

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Z-Based Confidence Interval

A common z-based confidence interval is:

CI = x̄ ± z × (σ / √n)*

when the population standard deviation is known.

In practical applications, analysts may use the sample standard deviation as an estimate of σ under appropriate conditions.

For smaller samples with an unknown population standard deviation, a t-distribution based confidence interval is generally preferred.

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T-Distribution vs Z-Distribution

Choosing between z and t methods is important.

Z-Based Interval

Uses a critical value from the standard normal distribution.

T-Based Interval

Uses a critical value from a t-distribution with degrees of freedom, typically:

df = n − 1

The t-distribution has heavier tails, especially with small sample sizes.

Calculator Pool’s current Confidence Interval Calculator uses a normal-approximation z critical value, so users should be aware of this limitation.

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When Should You Use a T-Interval?

A t-interval is commonly used when estimating a population mean from a sample when:

• The population standard deviation is unknown.
• The sample is relatively small.
• Conditions for the t-based method are reasonable.

For larger samples, the difference between z and t critical values becomes smaller.

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Why Does Calculator Pool Use a Z Critical Value?

The current calculator is designed as a simple normal-approximation confidence interval calculator.

This makes it easy to enter:

Mean

Standard Deviation

Sample Size

and:

Confidence Level

without requiring degrees of freedom or a t-distribution lookup.

For introductory calculations and suitable large-sample settings, the normal approximation can be useful.

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Confidence Interval for Small Samples

For a small sample, blindly using a z-based interval can be inappropriate when population standard deviation is unknown.

A t-based interval may be more suitable.

Therefore, when working with small samples, users should verify the assumptions and use the appropriate statistical method rather than relying solely on a generic confidence interval calculator.

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Confidence Interval for Large Samples

For large samples, the normal approximation often becomes more useful because the sampling distribution of the mean can be approximately normal under broad conditions.

However, sample size alone does not guarantee that every confidence interval method is appropriate.

The underlying data and sampling design still matter.

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Confidence Interval and Central Limit Theorem

The Central Limit Theorem is one reason normal-based methods are important in statistics.

Under suitable conditions, the distribution of sample means becomes approximately normal as sample size increases.

The standard error describes the spread of this sampling distribution.

Together, these ideas provide the foundation for many confidence interval calculations involving sample means.

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Confidence Interval and Sampling Distribution

A sampling distribution describes how a statistic would vary across repeated samples.

For sample means, the standard error describes the spread of the sampling distribution.

A confidence interval uses this estimated spread to create a range around the observed sample mean.

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Confidence Interval and Statistical Inference

Confidence intervals are a fundamental part of statistical inference.

They allow researchers to move beyond simply describing a sample.

For example:

Sample Mean = 50

by itself provides an estimate.

Adding:

95% CI = 46.08 to 53.92

provides additional information about uncertainty under the chosen method.

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Confidence Interval in Research

Researchers often report estimates together with confidence intervals.

For example:

Mean = 72.4

95% CI = 68.1–76.7

This communicates both the estimated mean and the uncertainty associated with it.

The correct confidence interval method depends on study design and statistical assumptions.

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Confidence Interval in Surveys

Confidence intervals are commonly associated with survey estimates.

For example, if a survey estimates an average measurement or proportion, the researcher may report an interval around the estimate.

However, real survey confidence intervals may need to account for:

• Sampling design
• Weights
• Clustering
• Nonresponse
• Design effects

Therefore, a simple calculator may not capture all complexities of a professional survey analysis.

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Confidence Interval in Medical Research

Confidence intervals are frequently used in medical and scientific research to communicate uncertainty around estimated effects or measurements.

For example, a study might report a mean difference together with a confidence interval.

The interpretation depends on the specific statistical model and study design.

A simple mean-based calculator should not be substituted for the appropriate analysis of a clinical study.

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Confidence Interval in Business Data

Businesses may use confidence intervals when estimating quantities such as:

• Average sales
• Mean customer spending
• Average delivery time
• Average production output

A confidence interval can provide additional context around a sample estimate.

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Confidence Interval in A/B Testing

A/B testing often involves comparing performance metrics between two groups.

Confidence intervals may be used to communicate uncertainty around estimated differences.

The exact calculation depends on whether the metric is:

• Mean
• Proportion
• Rate
• Difference
• Ratio
• Other statistic

Calculator Pool’s current tool is intended for the basic mean-based normal-approximation case.

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Confidence Interval for a Mean Example

Suppose a sample has:

Mean = 75

SD = 12

n = 36

For a 95% interval:

Standard Error

SE = 12 ÷ √36

SE = 2

Margin of Error

MOE = 1.96 × 2

≈ 3.92

Confidence Interval

75 − 3.92 = 71.08

75 + 3.92 = 78.92

Therefore:

95% CI ≈ 71.08 to 78.92

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How Confidence Level Changes the Result

Suppose the same sample has:

Mean = 75

SE = 2

90% Confidence Interval

Using:

z ≈ 1.645*

MOE:

≈ 3.29

Interval:

71.71 to 78.29

95% Confidence Interval

MOE:

≈ 3.92

Interval:

71.08 to 78.92

99% Confidence Interval

Using:

z ≈ 2.576*

MOE:

≈ 5.15

Interval:

69.85 to 80.15

As the confidence level increases, the interval becomes wider.

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Confidence Interval and Sample Size Example

Suppose:

Mean = 50

SD = 10

Sample Size = 25

SE:

2

Sample Size = 100

SE:

1

Sample Size = 400

SE:

0.5

At the same confidence level, the larger sample produces a narrower confidence interval.

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Why Confidence Intervals Are Useful

A confidence interval can communicate more information than a point estimate alone.

It shows:

• Estimated value
• Uncertainty
• Approximate precision
• Effect of sample size
• Effect of variability
• Effect of confidence level

This is why confidence intervals are widely used in statistical reporting.

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What Makes a Confidence Interval Wider?

An interval can become wider when:

Standard deviation increases

Sample size decreases

or:

Confidence level increases

Each of these can increase the margin of error.

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What Makes a Confidence Interval Narrower?

An interval can become narrower when:

Sample size increases

Standard deviation decreases

or:

Confidence level decreases

This follows directly from:

MOE = z × SD / √n*

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Confidence Interval Width Formula

The total width of a symmetric confidence interval is:

Upper Limit − Lower Limit

For:

46.08 to 53.92

the width is:

53.92 − 46.08 = 7.84

Since the interval is centered at 50:

Width = 2 × Margin of Error

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Confidence Interval and Precision

A narrower confidence interval is generally associated with greater precision of the estimate under the same statistical method.

For example:

95% CI: 49 to 51

is narrower than:

95% CI: 40 to 60

The first estimate has a smaller margin of error.

However, precision should not be confused with accuracy or freedom from bias.

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Confidence Interval and Accuracy

A narrow interval does not automatically mean the estimate is accurate.

If the data or sampling method is biased, a very precise estimate can still be systematically wrong.

Confidence intervals describe uncertainty under the assumptions of the statistical method; they do not automatically correct bad sampling or measurement.

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Common Confidence Interval Calculation Mistakes

Confusing Confidence Level With Probability of a Fixed Parameter

The conventional frequentist interpretation concerns long-run interval coverage.

Using the Wrong Critical Value

Different confidence levels require different critical values.

Confusing Standard Deviation With Standard Error

The confidence interval typically uses standard error:

SE = SD ÷ √n

not raw SD alone.

Ignoring Sample Size

Sample size affects the standard error.

Using a Z-Interval Automatically

For small samples with unknown population SD, a t-based interval may be more appropriate.

Treating the Interval as a Guarantee

A confidence interval does not guarantee that every calculated interval contains the true parameter.

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Confidence Interval Calculator vs Standard Error Calculator

These tools are closely related.

Standard Error Calculator

Calculates:

SE = SD ÷ √n

Confidence Interval Calculator

Uses SE together with a critical value to calculate:

Lower Limit

and:

Upper Limit

Therefore:

Standard Error → Margin of Error → Confidence Interval

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Confidence Interval Calculator vs Z-Score Calculator

A Z-Score Calculator answers:

How many standard deviations is this value from the mean?

A Confidence Interval Calculator answers:

What interval can be constructed around this sample estimate at a selected confidence level?

Both use concepts from the normal distribution, but their purposes are different.

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Confidence Interval Calculator vs Average Calculator

An Average Calculator determines the arithmetic mean of a dataset.

A Confidence Interval Calculator uses a sample mean as an input and adds an uncertainty range around that estimate.

Therefore:

Average → Point Estimate

Confidence Interval → Estimate + Uncertainty Range

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Confidence Interval Calculator vs Standard Deviation Calculator

A Standard Deviation Calculator measures the spread of individual observations.

A Confidence Interval Calculator uses standard deviation and sample size to estimate the uncertainty around the sample mean under the selected method.

This distinction is important in statistical analysis.

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Why Use an Online Confidence Interval Calculator?

Manual confidence interval calculations require:

• Standard error calculation
• Critical-value selection
• Margin-of-error calculation
• Lower-limit calculation
• Upper-limit calculation

An online confidence interval calculator can automate these steps.

It can help you:

• Calculate confidence intervals quickly
• Check statistics homework
• Compare confidence levels
• Understand margin of error
• Explore sample-size effects
• Reduce arithmetic mistakes

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Benefits of Calculator Pool’s Confidence Interval Calculator

Calculator Pool’s free confidence interval calculator online combines the main calculations in one place.

Enter:

Sample Mean

Standard Deviation

Sample Size

and:

Confidence Level

The calculator returns:

Confidence Interval

Standard Error

Margin of Error

Z Critical Value

Lower Limit

Upper Limit

This makes it convenient for basic statistical analysis and educational use.

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Frequently Asked Questions

What is a Confidence Interval Calculator?

A Confidence Interval Calculator calculates an estimated range around a sample statistic using a selected confidence level.

What is a confidence interval?

A confidence interval is a range calculated from sample data using a specified statistical procedure.

What is the confidence interval formula?

For the normal approximation used by Calculator Pool:

CI = Mean ± z × SE*

where:

SE = SD ÷ √n

What is the margin of error?

Margin of error is the distance from the sample estimate to the confidence-limit boundary.

MOE = z × SE*

What is standard error?

Standard error measures the sampling variability of an estimator such as a sample mean.

What is the difference between standard error and standard deviation?

Standard deviation describes variability among observations, while standard error describes sampling variability of an estimator.

What does 95% confidence interval mean?

It refers to the long-run coverage of the confidence-interval procedure: under repeated sampling and correct assumptions, about 95% of intervals constructed this way would contain the true parameter.

What does 90% confidence interval mean?

It refers to a procedure designed for approximately 90% long-run coverage under the relevant assumptions.

What does 99% confidence interval mean?

It refers to a procedure designed for approximately 99% long-run coverage under the relevant assumptions.

Why is a 99% confidence interval wider than a 95% confidence interval?

A higher confidence level requires a larger critical value, which increases the margin of error.

Does increasing sample size make a confidence interval narrower?

Generally yes, when other factors remain constant, because standard error decreases as sample size increases.

What happens if standard deviation increases?

The standard error and margin of error generally increase, making the confidence interval wider.

What happens if sample size decreases?

The standard error generally increases, which makes the interval wider.

What is the critical z-value?

It is the standard-normal cutoff associated with the selected confidence level.

What is the z-value for 95% confidence?

For a two-sided 95% normal approximation, the critical value is approximately:

1.96

What is the z-value for 90% confidence?

Approximately:

1.645

What is the z-value for 99% confidence?

Approximately:

2.576

What is the difference between z and t confidence intervals?

A z-based interval uses the standard normal distribution. A t-based interval uses a t-distribution and is commonly preferred for small samples when population standard deviation is unknown.

When should I use a t-distribution?

A t-based method is commonly appropriate for a mean when population standard deviation is unknown, particularly with smaller samples and suitable assumptions.

Does Calculator Pool use a t-distribution?

The current Calculator Pool Confidence Interval Calculator uses a normal-approximation z critical value, not a t-distribution.

Can I use this confidence interval calculator for small samples?

You can calculate a normal-approximation interval, but for small samples with unknown population SD, verify whether a t-based interval is more appropriate.

What is the lower confidence limit?

It is:

Mean − Margin of Error

for the symmetric interval used by the calculator.

What is the upper confidence limit?

It is:

Mean + Margin of Error

for the symmetric interval used by the calculator.

What is confidence interval width?

It is:

Upper Limit − Lower Limit

What is the relationship between confidence interval and standard error?

Standard error is multiplied by the critical value to obtain the margin of error.

What is the relationship between confidence interval and z-score?

A critical z-value is used to construct a normal-approximation confidence interval.

Is confidence interval the same as prediction interval?

No. A prediction interval concerns a future observation, while a confidence interval generally estimates an unknown population parameter.

Is confidence interval the same as margin of error?

No. Margin of error is only the distance from the estimate to one boundary. The confidence interval is the entire range.

Is confidence interval the same as standard deviation?

No. Standard deviation measures data variability, while confidence interval describes uncertainty around an estimate.

Can confidence intervals be negative?

Yes. The lower or upper limit can be negative depending on the data and parameter being estimated.

Can the confidence interval include zero?

Yes. Whether zero lies inside an interval can be important in statistical interpretation, depending on the parameter and research question.

Does a confidence interval prove a hypothesis?

No. Confidence intervals provide estimates and uncertainty information. Hypothesis-testing conclusions depend on the statistical framework being used.

Is a narrower confidence interval always better?

A narrower interval generally indicates greater precision under the same method, but precision is not the same as accuracy or absence of bias.

Can confidence intervals be used in research?

Yes. They are widely used to communicate uncertainty around statistical estimates.

Can confidence intervals be used for surveys?

Yes, although professional survey intervals may require methods that account for the actual sampling design.

Can I calculate a confidence interval from sample mean and standard deviation?

Yes, when the selected statistical method is appropriate. Calculator Pool requires those values plus sample size and confidence level.

Is the Confidence Interval Calculator free?

Yes. Calculator Pool’s Confidence Interval Calculator is free to use online.

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Related Calculators

You may also find these Calculator Pool tools useful:

Standard Error Calculator

Calculate standard error from standard deviation and sample size using:

SE = SD ÷ √n

Standard Deviation Calculator

Calculate population or sample standard deviation, variance, mean, minimum, and maximum.

Z-Score Calculator

Calculate how far a value is from the mean in standard-deviation units.

Average Calculator

Calculate the arithmetic mean, sum, count, minimum, and maximum.

Median Calculator

Find the middle value of a dataset after sorting.

Mode Calculator

Find the most frequently occurring value and its frequency.

Probability Calculator

Calculate basic probability from favorable and total possible outcomes.

Percentage Calculator

Perform common percentage calculations.

Permutation & Combination Calculator

Calculate nPr and nCr for counting and probability problems.

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Confidence Interval Calculator – Quick Summary

A Confidence Interval Calculator helps estimate a range around a sample mean at a selected confidence level.

Calculator Pool’s calculator uses:

Sample Mean

Standard Deviation

Sample Size

and:

Confidence Level

The main formula is:

CI = Mean ± z × SE*

where:

SE = SD ÷ √n

The margin of error is:

MOE = z × SE*

and the limits are:

Lower Limit = Mean − MOE

Upper Limit = Mean + MOE

For example:

Mean = 50

SD = 10

n = 25

Confidence Level = 95%

gives:

SE = 2

Margin of Error ≈ 3.92

95% CI ≈ 46.08 to 53.92

Higher confidence levels generally produce wider intervals, while larger sample sizes generally produce narrower intervals when other factors remain constant.

Confidence intervals are widely used in statistics, research, data analysis, surveys, science, economics, and business because they provide more information than a point estimate alone.

Calculator Pool’s free Confidence Interval Calculator online makes it easy to calculate the interval, standard error, margin of error, critical value, and confidence limits in one place.

For small samples where the population standard deviation is unknown, a t-distribution based confidence interval may be more appropriate than the normal approximation used by this calculator.