approximation

(noun)

An imprecise solution or result that is adequate for a defined purpose.

Related Terms

Examples of approximation in the following topics:

  • Linear Approximation

    • A linear approximation is an approximation of a general function using a linear function.
    • In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).
    • Linear approximations are widely used to solve (or approximate solutions to) equations.
    • Linear approximation is achieved by using Taylor's theorem to approximate the value of a function at a point.
    • If $f$ is concave-up, the approximation will be an underestimate.

  • The normal approximation breaks down on small intervals

    • Caution: The normal approximation may fail on small intervals The normal approximation to the binomial distribution tends to perform poorly when estimating the probability of a small range of counts, even when the conditions are met.
    • With such a large sample, we might be tempted to apply the normal approximation and use the range 69 to 71.
    • However, we would find that the binomial solution and the normal approximation notably differ:
    • TIP: Improving the accuracy of the normal approximation to the binomial distribution
    • The tip to add extra area when applying the normal approximation is most often useful when examining a range of observations.

  • Approximate Integration

    • Here, we will study a very simple approximation technique, called a trapezoidal rule.
    • The trapezoidal rule works by approximating the region under the graph of the function $f(x)$ as a trapezoid and calculating its area.
    • Although the method can adopt a nonuniform grid as well, this example used a uniform grid for the the approximation.
    • The function $f(x)$ (in blue) is approximated by a linear function (in red).
    • Use the trapezoidal rule to approximate the value of a definite integral

  • Introduction to evaluating the normal approximation

    • Many processes can be well approximated by the normal distribution.
    • While using a normal model can be extremely convenient and helpful, it is important to remember normality is always an approximation.

  • Conclusion

    • Many distributions in real life can be approximated using normal distribution.
    • If the graph is approximately bell-shaped and symmetric about the mean, you can usually assume normality.
    • The data are plotted against a theoretical normal distribution in such a way that the points form an approximate straight line .
    • We study the normal distribution extensively because many things in real life closely approximate the normal distribution, including:
    • The data points do not deviate far from the straight line, so we can assume the distribution is approximately normal.

  • Normal Approximation to the Binomial

    • This section shows how to compute these approximations.
    • The area in green in Figure 1 is an approximation of the probability of obtaining 8 heads.
    • For these parameters, the approximation is very accurate.
    • The accuracy of the approximation depends on the values of N and π.
    • Approximating the probability of 8 heads with the normal distribution

  • Optional Collaborative Classroom Activity

    • Construct an approximate 95% confidence interval for the true mean number of meals students eat out each week.
    • Construct the interval We say we are approximately 95% confident that the true average number of meals that students eat out in a week is between __________ and ___________.We say we are approximately 95% confident that the true average number of meals that students eat out in a week is between __________ and ___________.

  • Calculating a Normal Approximation

    • In this atom, we provide an example on how to compute a normal approximation for a binomial distribution.
    • The following is an example on how to compute a normal approximation for a binomial distribution.
    • The area in green in the figure is an approximation of the probability of obtaining 8 heads.
    • For these parameters, the approximation is very accurate.
    • Approximation for the probability of 8 heads with the normal distribution.

  • Evaluating the Normal approximation exercises

    • The superimposed normal curve approximates the distribution pretty well.
    • The data appear to be reasonably approximated by the normal distribution.

  • Newton's Method

    • Newton's Method is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • In numerical analysis, Newton's method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
    • Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is x0 - f(x0) / f'(x0).
    • This $x$-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated.
    • We see that $x_{n+1}$ is a better approximation than $x_n$ for the root $x$ of the function $f$.