Estimation is a technique for calculating information about a bigger group from a smaller sample, and statistics are crucial to analyzing data.
For instance, the average age of a city's population may be obtained by taking the age of a sample of 1,000 residents. While estimates aren't perfect, they are typically trustworthy enough to be of value.
Estimation in statistics involves using sample data to make educated guesses about a population's characteristics, such as mean, variance, or proportion. The population refers to the entire interest group, like all people in a country or all products made by a company.
Since it's often impractical to measure every member of a population, statisticians rely on samples to make inferences about the entire population.
Estimation helps to conclude population parameters based on sample data.
- Bias and Variance: Estimators can have bias, consistently overestimating or underestimating the true parameter. Variance measures the spread of estimator values around their predicted value. Both variance and bias impact the accuracy of estimators.
- Mean and Variance of Estimators: Estimators have the same mean and variance as random variables. The mean of an estimator should be equal to the parameter it is estimating. The variance of an estimator indicates its precision or variability.
In this article, we examine the significance of statistics, their function in the interpretation of data, and how efficient data analysis leads to the making of decisions based on accurate information.
Purpose of Estimation in Statistics
Statistical estimation is essential for finding unknown population parameters using sample data, like the mean and variance, without individual measurements.
- This evaluation is vital for decision-making in business and healthcare, informing strategies and treatment options.
- It is closely linked to hypothesis testing, contributing to scientific development, political decisions, public health, and economic choices.
- Risk assessment benefits from evaluation in managing probabilities and risk in finance and insurance.
- Quality control also relies on evaluation to ensure products and services meet standards by identifying and correcting deviations.
Types of Estimation
Estimation is of two types that include:
- Point Estimation
- Interval Estimation
Point Estimation
A single value (called a point estimate) is used to approximate an unknown population parameter. This single number, called a score estimator, gives a rough idea of the group's characteristics.
The population mean is estimated using the sample mean. Similar techniques can be applied to estimate other attributes, like percentages of specific characteristics in a population. While not always precise, these estimates offer a good understanding of the group's traits.
For instance, measuring the heights of random people can be used to estimate the average height of the entire group. If individuals measured were 5 feet, 6 feet, and 5 feet. We could estimate the average height to be around 5 feet.
Interval Estimation
Interval estimates give a range likely to contain the true parameter. This method recognizes data variability and estimation uncertainty.
When estimating the number of jelly beans in a jar, it is better to provide a range, known as a confidence interval, rather than a single guess. This range, such as 80 to 120 jelly beans, allows for uncertainty in the estimate and acknowledges the margin of error.
Confidence intervals give us a sense of freedom in our estimations, while point estimates only provide a single number without considering this uncertainty.
Confidence Interval in Interval Estimation
A confidence interval is the range of values, derived from a sample, that is likely to contain the true value of an unknown population parameter.
(1-α)% confidence interval means: "If we repeat the experiment many times, then (1 − α)% of the constructed intervals will contain the true parameter."
Example: A 95% confidence level means: "If we took 100 random samples and built 100 confidence intervals, we expect about 95 of them to contain the true population parameter."
Factors Affecting Estimation
Various factors affecting estimation are:
1. Sample Size: Larger sample sizes lead to more precise estimates, increasing the likelihood of accurately representing the population parameter.
- Estimating the average height of students in a school is more accurate with a larger sample size.
- Measuring just five students may not be reliable, but measuring 50 or even 500 students can provide a better idea of the true average height.
- A larger sample size leads to a more accurate estimate of the entire population's characteristics.
- In short, studying more individuals results in a more precise estimate of the entire population.
2. Sampling Method: The sampling method affects estimate accuracy. A random sample with every member having an equal chance ensures an unbiased estimate, improving accuracy.
- The sampling method is crucial for accurate estimations.
- Random sampling selects individuals purely by chance, giving each an equal chance of being chosen.
- This ensures a fair representation of the entire group, making it useful for determining distributions like colored candies in a jar or favorite ice cream flavors in a town without bias.
- Random sampling helps reflect the opinions of the whole group, not just a subset, leading to fair and unbiased findings for drawing accurate conclusions about a population or problem.
Estimation Methods
Several statistical techniques are used to estimate unknown parameters from data:
1. Method of Moments
- This method compares the moments (central tendency and spread) that are computed from the sample data to the corresponding moments in the population.
- The population parameters can be estimated by working out the resulting equations.
2. Maximum Likelihood Estimation (MLE)
- Maximum likelihood estimation (MLE) aims to find parameter values that give the highest chance of observing the given sample.
- It involves identifying values that maximize the likelihood of the observed data.
3. Least Squares Estimation
- LSE minimizes the sum of squared differences between observed values and predicted values.
- It’s widely used to find the best fit for a model, especially in regression, by reducing prediction errors.
4. Bayesian Estimation
- Bayesian Estimation updates prior belief about unknown parameters using observed data through Bayes' theorem.
- In Bayesian learning, parameters are random variables with probabilities assigned to them.
5. Interval Estimation
- This approach constructs a range of plausible values (e.g., confidence intervals) for an unknown parameter, reflecting uncertainty with a specified level of confidence.
Applications of Estimations in Computer Science
Machine Learning and AI
- Estimating parameters of models from data (e.g., weights in neural networks, coefficients in regression).
- Example: In linear regression, we use least squares estimation to find the line of best fit. In Bayesian networks, Bayesian estimation is used to compute posterior probabilities.
Data Compression
- Estimating the probability distribution of symbols for efficient encoding.
- Example: In Huffman coding, character frequencies are estimated to build optimal prefix codes. In arithmetic coding, probabilities are estimated in real time to achieve better compression ratios.
Cryptography
- Estimating the entropy of keys or messages to assess strength.
- Example: Estimating the key recovery success rate based on observed ciphertexts.
Computer Graphics and Vision
- Estimating depth, object boundaries, or motion in visual scenes.
- Example: In SLAM (Simultaneous Localization and Mapping), estimation is used to track a robot’s position in space.
Related Articles:
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice