Sampling distribution is essential in various aspects of real life, essential in inferential statistics. A sampling distribution represents the probability distribution of a statistic (such as the mean or standard deviation) that is calculated from multiple samples of a population. It helps us to understand how a statistic varies across different samples and is crucial for making inferences about the population.
Sampling distribution is the probability distribution of a statistic based on random samples of a given population. It is also know as finite distribution.
In this article, we will discuss the Sampling Distribution in detail and its types, along with examples, and go through some practice questions, too.
Important Terminologies in Sampling Distribution
Some important terminologies related to sampling distribution are given below:
- Statistic: Summary value from a sample (e.g., mean, median).
- Parameter: Summary value from a population.
- Sample: A subset of a population.
- Population: The entire group being studied.
- Sampling Distribution: Distribution of a statistic across many samples.
- Central Limit Theorem (CLT): Sample means follow a normal distribution as the sample size increases.
- Standard Error: Standard deviation of the sampling distribution.
- Bias: Systematic error causing deviation from the true value.
- Confidence Interval: Range likely to contain the population parameter.
- Sampling Method: How samples are chosen (random, stratified, etc.).
- Inferential Statistics: Concluding a population from samples.
- Hypothesis Testing: Making decisions about population parameters using sample data.
Factors Influencing Sampling Distribution
The variability of a sampling distribution is measured by standard error or population variance, depending on the context and the type of inference required. Both measure how spread out the data is around the mean.
Main factors influencing the variability of a sampling distribution are:
- Number Observed in a Population: The symbol for this variable is "N." It is the measure of observed activity in a given group of data.
- Number Observed in Sample: The symbol for this variable is "n." It is the measure of observed activity in a random sample of data that is part of the larger grouping.
- Method of Choosing Sample: How you chose the samples can account for variability in some cases.
Types of Distributions
3 main types of sampling distributions are:
- Sampling Distribution of Mean
- Sampling Distribution of Proportion
- T-Distribution
Sampling Distribution of Mean
The sampling distribution of the mean refers to the probability distribution of sample means that you get by repeatedly taking samples (of the same size) from a population and calculating the mean of each sample.
Key concepts of Sampling Distribution of Mean
- Population Mean (μ): The average of the entire population.
- Sample Mean (x̄): The average of a sample taken from the population.
- Sampling Distribution of the Mean: If you take multiple samples and plot their means, that plot will form the sampling distribution of the mean.
For any population with mean µ and standard deviation σ:
- Mean, or center of the sampling distribution of x̄, is equal to the population mean, µ.
µ_{x^{-}} = µ
There is no tendency for a sample mean to fall systematically above or below µ, even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution is an unbiased estimate of the population mean µ.
- Standard deviation of the sampling distribution is σ/√n, where n is the sample size.
σ_x = σ/√n
Where
- σ_x - Standard Deviation of Sampling Deviation
- σ - Population Standard Deviation
- n - Sample size
sampling distribution of standard deviation
Sampling Distribution of Proportion
Sampling distribution of a proportion focuses on proportions in a population. Here, you select samples and calculate their corresponding proportions. The means of the sample proportions from each group represent the proportion of the entire population.
Sampling distribution of proportion - 1
Sampling distribution of proportion - 2The formula for the sampling distribution of a proportion (often denoted as p̂) is:
p̂ = x/n
Where:
- p̂ is the Sample Proportion
- x is the Number of "successes" or occurrences of the Event of Interest in the Sample
- n is Sample Size
This formula calculates the proportion of occurrences of a certain event (e.g., success, positive outcome) within a sample.
T-Distribution
Sampling distribution involves a small population or a population about which you don't know much. It is used to estimate the mean of the population and other statistics such as confidence intervals, statistical differences, and linear regression. T-distribution uses a t-score to evaluate data that wouldn't be appropriate for a normal distribution.
The formula for the t-score, denoted as t, is:
t = [x - μ] / [s /√(n)]
Where:
- x is the Sample Mean
- μ is Population Mean (or an estimate of it)
- s is the Sample Standard Deviation
- n is Sample Size
This formula calculates the difference between the sample mean and the population mean, scaled by the standard error of the sample mean. The t-score helps to assess whether the observed difference between the sample and population means is statistically significant.
Solved Examples of Sampling Distribution
Example 1: Mean and standard deviation of the tax value of all vehicles registered in a certain state are μ=$13,525 and σ=$4,180. Suppose random samples of size 100 are drawn from the population of vehicles.
Find
- mean μx̄
- standard deviation σx̄ of the sample mean x̄
Solution:
Since n = 100, the formulas yield
μx̄ = μ = $13,525
σx̄ = σ / √n = $4180 / √100
σx̄ = $418
Example 2: A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 36,000 miles. Assume that the distribution of lifetimes of such tires is normal.
Solution:
Here, we will assume and use units of thousands of miles.
Then sample mean x̄ has
- Mean: μx̄ = μ = 38.5
- Standard Deviation: σx̄ = σ/√n = 2.5/√5 = 1.11803
Since the population is normally distributed, so is x̄, hence,
P (X < 36) = P(Z < {36 - μx̄}/σx̄)
P (X < 36) = P(Z < {36 - 38.5}/1.11803)
P (X < 36) = P(Z < -2.24)
P(X < 36) = 0.0125
Therefore, if the tires perform as designed then there is only about a 1.25% chance that the average of a sample of this size would be so low.
Practice Questions on Sample Distribution
Question 1: Random samples of size 225 are drawn from a population with a mean of 100 and a standard deviation of 20. Find the mean and standard deviation of the sample mean.
Question 2: Random samples of size 64 are drawn from a population with a mean of 32 and a standard deviation of 5. Find the mean and standard deviation of the sample mean.
Question 3: A population has a mean of 75 and a standard deviation of 12.
- Random samples of size 121 are taken. Find the mean and standard deviation of the sample mean.
- How would the answers to part
- Change if the size of the samples were 400 instead of 121?
Question 4: A population has a mean of 5.75 and a standard deviation of 1.02.
- Random samples of size 81 are taken. Find the mean and standard deviation of the sample mean.
- How would the answers to part
- Change if the size of the samples were 25 instead of 81?
Question 5: The Numerical population of grade point averages at a college has a mean of 2.61 and a standard deviation of 0.5. If a random sample of size 100 is taken from the population, what is the probability that the sample mean will be between 2.51 and 2.71?
Question 6: Random samples of size 1,600 are drawn from a population in which the proportion with the characteristic of interest is 0.05. Decide whether or not the sample size is large enough to assume that the sample proportion is normally distributed.
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