Empirical Probability Examples

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Introduction to Empirical Probability

An empirical probability (also called experimental probability) is calculated by collecting data from past trials of the experiments. The experimental probability obtained is used to predict the future likelihood of the event occurring.

Formula and Examples Empirical/ Experimental Probability

To calculate an empirical/ experimental probability, one can use the formula

$$P(A)=\frac{\text{Number of trials in which $A$ occurs} }{$\text{Total number of trials}}$$

  • Coin Flip: Let us flip a coin 200 times and get heads 105 times. The empirical probability of getting heads is $\frac{105}{200} = 0.525%, or 52.5%.
  • Weather Prediction: Let you track the weather for a month and see that it rained 12 out of 30 days. The empirical probability of rain on a given day that month is $\frac{12}{30} = 0.4$ or 40%.
  • Plant Growth: Let you plant 50 seeds and 35 sprout into seedlings. The experimental probability of a seed sprouting is $\frac{35}{50} = 0.70$ or 70%.
  • Board Game: Suppose you play a new board game 10 times and win 6 times. The empirical probability of winning the game is $\frac{6}{10} = 0.6$ or 60%.
  • Customer Preferences: In a survey of 100 customers, 80 prefer chocolate chip cookies over oatmeal raisins. The empirical probability of a customer preferring chocolate chip cookies is $\frac{80}{100} = 0.80$ or 80%.
  • Basketball Game: A basketball player practices free throws and makes 18 out of 25 attempts. The experimental probability of the player making their next free throw is $\frac{18}{25} = 0.72$ or 72%.

Empirical Probability From Frequency Tables

A frequency table calculates the probability that a certain data value falls into any data group or class. Consider the frequency table of examination scores in a certain class.

ClassFrequency ($f$)$frf$
40 – 491$\frac{1}{20}=0.05$
50 – 592$\frac{1}{20}=0.10$
60 – 693$\frac{3}{20}=0.15$
70 – 794$\frac{4}{20}=0.20$
80 – 896$\frac{6}{20}=0.30$
90 – 994$\frac{4}{20}=0.20$

Let event $A$ be the event that a student scores between 90 and 99 on the exam, then

$$P(A) = \frac{\text{Number of students scoring 90-99}}{\text{Total number of students}} = \frac{4}{20} = 0.20$$

Notice that $P(A)$ is the relative frequency of the class 90-99.

Empirical Probability and Classical Probability

Key Points Empirical/ Experimental Probability

  • It is based on actual data, not theoretical models.
  • It is a good approach when the data is from similar events in the past.
  • The more data you have, the more accurate the estimate will be.
  • It is not always perfect, as past results do not guarantee future outcomes.

Limitations Empirical/ Experimental Probability

  • It can be time-consuming and expensive to collect enough data.
  • It may not be representative of the future, especially if the underlying conditions change.

FAQS about Empirical/ Experimental Probability

  1. Define empirical probability.
  2. How one can compute empirical probability, write the formula of empirical probability.
  3. Give real-life examples of empirical/ experimental probability.
  4. What are the limitations of empirical/ experimental probability?
  5. How does empirical/ experimental probability resemble with frequency distribution, explain.
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Median of Ungrouped Data

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Introduction to Median of Ungrouped Data

The post is about calculating the median ungrouped data. The median is the most central point (middlemost central value) of the data/set of observations, with the condition that the data or set of observations should be arranged in ascending or descending order. The median divides the data into two equal parts. That is the main objective of the median.

It is important to note that the criteria for finding the median for grouped and ungrouped data are different.

The primary and secondary data can be defined as:

  1. Primary data, also called raw or ungrouped data, does not undergo any statistical procedure/method, which is not in the form of frequency distribution.
  2. Secondary data may also be called group data if it is in the form of frequency distribution.

Let us discuss how to find the median for ungrouped data.

There are two cases for ungrouped data. These cases are based on no of observations which is $n$

When the number of observations is odd (Say $n$ i.e. $n$ is odd), and when the number of observations is even (Say $n$ i.e. $n$ is even).

Median Calculations

The data below contains the odd number of observations.

Observation No.
(Ascending Order)		1st2nd3rd4th5th6th7th8th9th10th11th
Data Values81899096100102103104108109118
(Descending Order)1110987654321

Since the number of observations is odd ($n = 11$), the central value after arranging in ascending order will be the 6th value. and the 6th value is 102. That is the median is 102 for the above data.

The position of the median can be located mathematically, as follows:

\begin{align*}
\tilde{x} &= \left( \frac{n+1}{2} \right)th\,\, \text{value}\\
&=\frac{11+1}{2} = 6th\,\, \text{value}
\end{align*}

The value at the 6th position (from sorted data) is 102. The $\tilde{x}$ can be read as “x-tild” which is the notation of the median.

Median for Even Numbers of Observations

Consider the following data that contains an even number of observations.

Observation No.12345678910
Data Values81100961089010210410310989

Data after sorting (either in ascending or descending order) is

Observations No.1st2nd3rd4th5th6th7th8th9th10th
x81899096100102103104108109

Since $n=10$ which is even, the central position (that is median) lies between the 5th value and the 6th value. This central value is the average of the 5th and 6th values (from the sorted data). The average of these two central observations is called the median. The two central positions are 100 and 102, take the average of these two numbers and find the median.

$$Median = \frac{100+102}{2} = 101$$

Median Formula for Large Data Sets

The median formula for large or small data sets can be represented mathematically.

  • For large data sets one can find the median of data mathematically. The formula for both odd number of observations and even numbers of observations is different.

The point to remember when computing the median is that

  • For an odd number of observations, the median is the centermost value after sorting the data
  • For an even number of observations, the median is the average of two central values after sorting the data

\begin{align*}
\tilde{x} &= \frac{1}{2} \left[ \left(\frac{n}{2}th \, \, value \right)+ \left(\frac{n}{2}+1 \right)the \,\, value \right]\quad \quad \text{(When observations are even)}\\
&= \frac{n+1}{2} \quad \quad \text{(when observations are odd)}
\end{align*}

The other way of the median formula is

Median of ungrouped data

Consider, a data set containing 157 observations. To compute the median, first of all, you need to sort the data in either ascending or descending order. The formula for this data will be

$$\tilde{x} = \frac{n+1}{2} = \frac{157+1}{2}=79th$$.

The 79th observation in the sorted data will be the median of the data.

In case, if there are even number of observations (say $n=396$, the median will be

\begin{align*}
\tilde{x} &= \frac{1}{2}\left[\left(\frac{n}{2}\right)th + \left(\frac{n+1}{2}\right)th \right]\\
&=\frac{1}{2} \left[\frac{396}{2}th + \frac{396}{2}+1 \right]\\
&= \frac{1}{2} \left[198th + 199th\right]
\end{align*}

The average of 198th value and 199th value from the sorted data will be the median of the data.

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Statistical Inference: An Introduction

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Introduction to Statistical Inference

Inference means conclusion. When we discuss statistical inference, it is the branch of Statistics that deals with the methods to make conclusions (inferences) about a population (called reference population or target population), based on sample information. The statistical inference is also known as inferential statistics. As we know, there are two branches of Statistics: descriptive and inferential.

Statistical inference is a cornerstone of many fields of life. It allows the researchers to make informed decisions based on data, even when they can not study the entire population of interest. The statistical inference has two fields of study:

Statistical Inference

Estimation

Estimation is the procedure by which we obtain an estimate of the true but unknown value of a population parameter by using the sample information that is taken from that population. For example, we can find the mean of a population by computing the mean of a sample drawn from that population.

Estimator

The estimator is a statistic (Rule or formula) whose calculated values are used to estimate (a wise guess from data information) is used to estimate a population parameter $\theta$.

Estimate

An estimate is a particular realization of an estimator $\hat{\theta}$. It is the notation of a sample statistic.

Types of Estimators

An estimator can be classified either as a point estimate or an interval estimate.

Point Estimate

A point estimate is a single number that can be regarded as the most plausible value of the $\theta$ (notation for a population parameter).

Interval Estimate

An interval estimate is a set of values indicating confidence that the interval will contain the true value of the population parameter $\theta$.

Testing of Hypothesis

Testing of Hypothesis is a procedure that enables us to decide, based on information obtained by sampling procedure whether to accept or reject a specific statement or hypothesis regarding the value of a parameter in a Statistical problem.

Note that since we rely on samples, there is always some chance our inferences are not perfect. Statistical inference acknowledges this by incorporating concepts like probability and confidence intervals. These help us quantify the uncertainty in our estimates and test results.

Important Considerations about Testing of Hypothesis

  • Hypothesis testing does not prove anything; it provides evidence for or against a claim.
  • There is always a chance of making errors (Type I or Type II).
  • The results are specific to the chosen sample and significance level.

Statistical Inference in Real-Life

Some real-life examples of inferential statistics:

  1. Medical Trials: When a new drug is developed, it is tested on a sample of patients to infer its effectiveness and safety for the general population. Statistical inference helps determine whether the observed effects are due to the drug or random chance.
  2. Market Research: Companies use inferential statistics to understand consumer preferences and behaviours. By surveying a sample of consumers, they can infer the preferences of the broader market and make informed decisions about product development and marketing strategies.
  3. Public Health: Epidemiologists use statistical inference to track the spread of diseases and the effectiveness of interventions. Analyzing sample data one can infer the overall impact of a disease and the effectiveness of measures like vaccinations.
  4. Quality Control: Manufacturers use statistical inference to monitor product quality. By sampling a few items from a production batch, they can infer the quality of the entire batch and make decisions about whether to continue production or make adjustments.
  5. Election Polling: Pollsters use samples of voter opinions to infer the likely outcome of an election. Statistical inference helps estimate the proportion of the population that supports each candidate and the margin of error in these estimates.
  6. Education: Educators and policymakers use statistical inference to evaluate the effectiveness of teaching methods and educational programs. By analyzing test scores and other performance metrics from a sample of students, they can infer the impact of these methods on the broader student population.
  7. Environmental Studies: Researchers use statistical inference to assess environmental impacts. For example, by sampling air or water quality in specific locations, they can infer the overall environmental conditions and the effectiveness of pollution control measures.
  8. Sports Analytics: Teams and coaches use statistical inference to evaluate player performance and strategy effectiveness. By analyzing data from a sample of games, they can infer the overall performance trends and make decisions about training and game strategy.
  9. Finance: Investors and financial analysts use statistical inference to make decisions about investments. By analyzing sampled historical data of stocks or other financial instruments, one can infer future performance and make informed investment decisions.
  10. Customer Satisfaction: Businesses use statistical inference to gauge customer satisfaction and loyalty. By surveying a sample of customers, one can infer the overall satisfaction levels and identify areas for improvement.

FAQs about Statistical Inference

  1. Define the term estimation.
  2. Define the term estimate.
  3. Define the term estimator.
  4. Write a short note on statistical inference.
  5. What is statistical hypothesis testing?
  6. What is the estimation in statistics?
  7. What are the types of estimations?
  8. Write about point estimation and intervention estimation.

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