Mean or Expected Value and Standard Deviation Introductory Statistics

In simpler terms, it’s essentially what you can expect to happen on average after many trials of an event. This powerful tool offers predictive insights into future events by considering all possible outcomes and their probabilities. An insurance company would like to determine the proportion of all medical doctors
who have been involved in one or more malpractice lawsuits.

Very informally, many ratio scales can be described as specifying “how much” of something (i.e. an amount or magnitude). Ratio scale is often used to express an order of magnitude such as for temperature which among the central tendency measurements is used best for the variable “lottery ticket”? in Orders of magnitude (temperature). The ordinal type allows for rank order (1st, 2nd, 3rd, etc.) by which data can be sorted but still does not allow for a relative degree of difference between them.

This would provide an expected sales figure that can serve as a KPI. This method allows us to model complex systems with multiple variables and uncertain outcomes, providing a range of potential results rather than a single deterministic value. Expected Value (EV) in Monte Carlo simulations is applied by running numerous trials and averaging the outcomes. Each trial represents a possible scenario, with its outcome weighted by its probability. The EV is calculated as the sum of all possible values each multiplied by their respective probabilities. In terms of expected value, we aim for low bias and low variance as it provides the most accurate model predictions.

  1. The company selects [latex]500[/latex] doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.
  2. The first investment, a software company, has a 10% chance of returning 💲5,000,000 profit, a 30% chance of returning 💲1,000,000 profit, and a 60% chance of losing the million dollars.
  3. To find the standard deviation, add the entries in the column labeled (x – μ)2P(x) and take the square root.
  4. For example, the mean may not work well with quantitative datasets that contain extremely large or extremely small values.
  5. The sample mean is used as the ‘best guess’ approximation of the population mean.

These functions take into account not just the monetary outcome, but also factors like risk tolerance and opportunity cost. Moreover, in decision trees, EV aids in selecting the best split at each node by calculating the expected information gain or reduction in entropy. Prepare for your next interview with our insightful article on Expected Value Interview questions and answers. Gain an understanding of key concepts and improve your problem-solving skills in this important area of probability and statistics.

The technical term is “arithmetic mean,” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.” Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer.

In this function, probabilities and outcomes are arrays representing the probability distribution and corresponding outcomes respectively. We use np.multiply() to multiply each outcome with its corresponding probability. Then we sum up these products using np.sum(), which gives us the Expected Value. In the context of KPIs, EV can help quantify potential outcomes and their likelihoods for various business scenarios. For instance, if a company is considering launching a new product, they could use EV to estimate sales figures by multiplying each possible sales scenario by its probability of occurrence.

Ratio scale

Thus, despite the higher price, the lower probability makes the second option less profitable on average. In essence, the connection between EV and CLT lies in the fact that as sample size increases, the sample mean converges towards the population mean or EV. Thus, the CLT provides a theoretical underpinning for why we can use the EV as an estimate of the long-term average in real-world applications. The sample is the 500 doctors selected at random from the professional directory. In this post, I shared how I analyzed the past lottery data statistically to decide to buy a lottery ticket.

The fraction is equal to 0.498 which is very close to 0.5, the expected probability. The statistic is an estimate of a population parameter, in this case the mean. The theory of probability began with the study of games of chance such as poker. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities.

Level of measurement

Among unbiased estimators, there often exists one with the lowest variance, called the minimum variance unbiased estimator (MVUE). A desired property for estimators is the unbiased trait where an estimator is shown to have no systematic tendency to produce estimates larger or smaller than the provided probability. Additionally, unbiased estimators with smaller variances are preferred over larger variances because it will be closer to the “true” value of the parameter. The unbiased estimator with the smallest variance is known as the Minimum-variance unbiased estimator(MVUE). The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample.

The company selects [latex]500[/latex] doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit. Expected value and variance are two fundamental concepts in statistics, both providing different insights about the probability distribution of a random variable. Expected value is essentially the mean of a random variable’s possible outcomes weighted by their respective probabilities. It gives us an idea of what to expect on average from numerous trials. The ordinal scale places events in order, but there is no attempt to make the intervals of the scale equal in terms of some rule. Rank orders represent ordinal scales and are frequently used in research relating to qualitative phenomena.


Most people will lose their money, making the expected value misleading. It also ignores non-monetary factors like the thrill of playing or the despair from losing repeatedly. For instance, in financial investment decisions, it helps determine whether the potential return on an investment outweighs the risks involved. By calculating the EV of different investment options, one can choose the most profitable option with acceptable risk levels. The expected value is a key concept in the Law of Large Numbers, which states that as the number of trials or observations increases, the actual result will converge to the expected result.

For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times.

For some probability distributions, there are short-cut formulas for calculating μ and σ. The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median.

Watch the following video for a brief introduction to statistics. A “friend” offers you the following “deal.” For a 💲10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson.

Ratios are not meaningful since 20 °C cannot be said to be “twice as hot” as 10 °C (unlike temperature in Kelvins), nor can multiplication/division be carried out between any two dates directly. Interval type variables are sometimes also called “scaled variables”, but the formal mathematical term is an affine space (in this case an affine line). We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey [latex]100[/latex] families with children in the school. Three of the families spent [latex]$65[/latex], [latex]$75[/latex], and [latex]$95[/latex], respectively. Like data, probability distributions have standard deviations.

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