Statistics Interview Questions and Answers for 2025

For univariate analysis of a numerical variable, the must use visualizations are histograms and box and whisker plot (or box plot). Scatter plots are used to perform multivariate analysis of numerical variables. 

• Histograms 

A histogram is a graphic representation of the distribution of data that has been grouped into classes. It is a type of frequency chart that is made up of a number of rectangles. Each piece of data is sorted, then each value is assigned to the proper class interval. The frequency of each class interval is determined by the number of data values that fall within it. A specific class of data is represented by each rectangle in the histogram. The width of the rectangle represents the width of the class. It is commonly used to determine 

• Box and whisker plot (box plot) 

A box plot shows the maximum and minimum values, the first and third quartiles, and the median value, which is a measure of central tendency. In addition to these quantities, it also explains the symmetry and variability of the data distribution. Outliers in the dataset are frequently visualised using this visualization. 

• Scatter plot 

The scatterplot is a very helpful and effective tool that is frequently used in regression analysis. A pair of observed values for the dependent and independent variables are represented by each point. Before selecting a suitable model, it enables graphically determining whether a relationship between two variables exists. These scatterplots are also very helpful for residual analysis because they let you check whether the model is a good fit or not.

Covariance and correlation coefficient reveals the relationship and the strength of relationship between the two variables.  

Covariance is a measure of how two random variables in a data set will change jointly. When two variables are positively correlated and moving in the same direction, this is referred to as positive covariance. A negative covariance denotes an inverse relationship between the variables or a movement in the opposite directions. For instance, a student's performance on a particular examination improves with increased attendance, which is a positive correlation, whereas a decrease in demand caused by a rise in the price of an item is a negative correlation. When the covariance value is zero, the variables are said to be independent of one another and have no influence on one another. If the covariance value is higher than 0, it means that the variables are positively correlated and move in the same direction. The variables are negatively correlated and move in the opposite direction when the correlations have a negative value. 

Similar information is given by the correlation coefficient and the covariance. The fact that the correlation coefficient will always retain a value between negative one and one is its benefit over covariance. A perfect positive correlation exists between the variables under study when the correlation coefficient is 1. In other words, as one moves, the other follows suit proportionally in the opposite direction. A less than perfect positive correlation is present if the correlation coefficient is less than one but still larger than zero. The correlation between the two variables is stronger as the correlation coefficient approaches one. There is no observable relationship between the variables when the correlation coefficient is zero. That means it is difficult to predict the movement of the other variable if one variable moves. The variables are perfectly negatively or inversely connected if the correlation coefficient is zero, or negative one. One variable will drop proportionally in response to an increase in the other. The variables will oscillate in opposing directions. If the correlation coefficient is more than negative one, it means that the negative correlation is not perfect. The correlation increases as it gets closer to being negative one.  

Think about a sample taken from a population as a whole with a mean value. It's possible that we'll obtain an entirely different mean if we take another sample from the same population. Let's say you gathered ten distinct samples. You'll observe that the sample mean is influenced by the members belonging to their own sample. Hence, using just one value is not the best course of action.

A fresh dataset of sample means is produced by the new samples that were collected. There is a certain distribution of these values. The phrase "sampling distribution" is used to describe a distribution made out of samples. We are dealing with a sampling distribution of the mean in this instance. These values are distinct when we look at them closely, but they are centred on one particular value.

Every sample mean in this analysis approximates the population mean. The value they centre on may provide a very accurate indication of the population mean. In fact, we anticipate getting a pretty accurate approximation of the population mean if we take the average of those sample means. We see a normal distribution when we visualise the distribution of the sampling means and the Central Limit confirms that. The sampling distribution of the mean will resemble a normal distribution regardless of the underlying population distribution, whether it be binomial, exponential, or another type.

As a result, even when the population is not normally distributed, we can still conduct tests, work through issues, and draw conclusions using the normal distribution according to the central limit theorem.

There are various performance measures or metrics that can help to evaluate the performance of a classification model. However, it depends on the kind of problem we are dealing it. At times, accuracy might not be a good idea for evaluation and we need to focus on certain aspects of the results rather than the accuracy as a whole. The most common metrics used for the purpose are –

• Confusion matrix

A confusion matrix is one of the evaluation techniques for machine learning models in which you compare the results of all the predicted and actual values. Confusion matrix helps us to derive several different metrics for evaluation purpose such as accuracy, precision, recall, and F1 score which are widely used across different classification use cases.

• ROC AUC curve

The probability curve, the Receiver Operator Characteristic (ROC) separates the signal from the noise by plotting the True Positive Rate (TPR) versus the False Positive Positive Rate (FPR) at different threshold values. A classifier's capacity to distinguish between classes is measured by the Area Under the Curve (AUC). The performance of the model at various thresholds between positive and negative classes is improved by a higher AUC. The classifier can correctly discriminate between all Positive and Negative class points when AUC is equal to 1. The classifier would be predicting all negatives as positives and vice versa when AUC is equal to 0.

• Jaccard index

Jaccard Index or also known as Jaccard similarity coefficient. If y is the actual label and ŷ is the predicted value then we can define Jaccard index as the size of the intersection by the size of the union of two labelled sets.

Consider if you have a total of 50 observations, out of which your model predicts 41 of them correctly, then the Jaccard index is given as 41 / (50 + 50 - 41) = 0.69. The Jaccard index of 0.69 defines that the model predicts on the test set with an accuracy of 69%. So, a Jaccard index ranges from 0 to 1 where an index value of 1 implies maximum accuracy.

• Log loss

Log loss or logarithmic loss measures the performance of a classifier where predicted output is a probability value between 0 and 1. We can calculate the log loss using the log loss equation which measures how far each prediction is from the actual label. It is obvious that most ideal classifiers have a lower value of log loss. So the classifier with lower log loss has better accuracy.

Confusion matrix is one of the evaluation methods for machine learning models that compares the outcomes of all the expected and actual values.

The figure representing confusion matrix has four different cases: 

1. There are five instances where the predicted value and the actual value are both true. This is referred to as a True Positive case, where True denotes that the values are identical (true and true) and Positive denotes that the situation is true. Example: A diabetes test is positive for a diabetic patient. 
2. There are four instances where both the predicted value and the actual value are false. This is referred to as a True Negative situation, where True denotes identical numbers (false and false) and Negative denotes a negative outcome. Example: A diabetes test is negative for a non-diabetic patient. 
3. In three instances, the projected value is true, but the actual value is false. False denotes that the values are different (false and true), while Positive means that the predicted value is positive. This is referred to as a False Positive event. Example: A diabetes test is positive for a non-diabetic patient. 
4. There are two situations where the projected value is false, whereas the actual value is true. This situation is known as a False Negative Case, where False denotes that the values (true and false) are different, and Negative denotes that the predicted value is negative. Example: A diabetes test is negative for a diabetic patient. 

In this matrix, the values in green are correctly identified by the model and the values in red are wrongly identified by the model. Confusion matrix can also be used for non-binary target variables. 

The occurrence of Type 1 Error, also known as a False Positive event, occurs when the expected value is positive but it is actually negative. When the actual value is positive when the predicted value is negative, this is known as a False Negative event and results in Type 2 Error. For instance, if we consider rain to be a positive event, then your device's prediction that it would rain today but it didn't actually happen is a type 1 error, while your device's prediction that it wouldn't rain today but it actually did happen is a type 2 error.

The performance of a regression model is evaluated based on how close or far the predictions are to the actual value. Primarily, there are three metrics widely used to evaluate the performance of regression tasks, namely, Mean Absolute Percentage Error (MAPE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE). All three of these metrics evaluate the distance between the predictions and the actual values. MAPE is generally used in tasks where there is presence of outliers, like the timeseries data. MSE and RMSE is used when the outliers are not present or relatively less. RMSE is generally preferred over MSE as it provides the metric value in the same unit as the target variable which makes it easy for comparison.

The R2 score, often known as the R-squared value, indicates how much of the overall variability can be explained by the regression. It is a relative scale with values ranging from 0 to 1. An R2 score of 0 indicates that our regression model does not explain any of the data variability. An R2 value of 1 indicates that our regression model completely accounts for the data's variability, which is uncommon or practically impossible. The degree of topic complexity and the number of variables employed both affect the R2 score. However, the higher the score, the better it is. However, R2 score has a drawback because it is a monotonically growing function. This implies that the value of the R2 score will increase each time we include a new variable in the regression model, giving the impression that the more variables we include, the better our model will be. This isn't always the case because the additional variable might not have much of an impact on the model. To take care of this, an adjusted R2 score is preferred which penalises the model for using an insignificant variable. This will ensure that the score is higher only if we have used significant variables and have avoided the insignificant ones.

Also known as the F1-Score or the F-Score, the F-Measure is a numerical value. It evaluates how accurate a test is. In a perfect scenario, both the precision and recall values would be high. However, there is always a trade-off between recall and precision, and unfortunately, we must prioritise one over the other. The two components of the F1 score are precision and recall. The F1 score aims to combine the precision and recall measures into a single metric. This F-score is what we use to compare two models. In terms of the formula, F1 score is the harmonic mean of precision and recall and given by –  

The value of F1 score ranges between 0 and 1. An F1 score of 1 is regarded as ideal, whereas a score of 0 indicates that the model is a complete failure.  

The sum of the square differences between the observed dependent variable and its mean is known as the sum of square total (SST). It is a measurement of the dataset's overall variability.

The sum of the squares between the predicted value and the dependent variable's mean is known as the sum of squares due to regression (SSR). It explains how well the data fit our regression line. If this value is the same as the SST, our regression model perfectly captures the observed variability.

The difference between the actual value and the predicted value is known as the Sum of Squared Error (SSE). Usually, we wish to reduce the error. The regression's estimating power increases with decreasing error.  

The overall variability of the data set provided by SST is equal to the sum of the variability described by the regression line, or SSR, and the unexplained variability, or SSE.

A regression model can be assessed using MAPE, MSE, and RMSE metrics. In statistics, the Mean Absolute Percentage Error, or MAPE, is a measure of forecasting method accuracy. The accuracy is typically expressed as a ratio. If there are several outliers present, it is usually preferred and known as the average absolute deviation. Two vectors—the vector of predictions and the vector of target values—can be measured using the Mean Absolute Error (MAE) and the Root Mean Squared Error (RMSE), respectively. The l2 and l1 norms, respectively, relate to the calculation of the RMSE and MAE. The RMSE works exceptionally well and is typically preferred when the outliers are exceedingly rare, as in a bell-shaped curve.

Distance is a measurement of how far apart two objects are. Therefore, a distance is equivalent to a real number. If both objects are the same, it is zero; otherwise, it is positive. The distance units that are most frequently used are the Euclidean, Manhattan, Minkowski, and Hamming. The Manhattan distance and the Euclidean distance can both be thought of as generalisations of the Minkowski distance. The Minkowski distance for p = 1 is also known as the Manhattan distance, the L1 norm, or the absolute distance. The Euclidean distance is referred to as the distance when p = 2.

One common way to identify outliers is through the internal limits or whiskers derived from the interquartile range. The interquartile range represents the 50% of the observations. The median is located at the centre of this range. When there are minimum or maximum values at the extreme sides then we can define the cut-off value to determine the outliers using the formula –

• Lower limit = First quartile – (1.5 x interquartile range)
• Upper limit = Third quartile – (1.5 x interquartile range)

These two data values are known as adjacent points. If we find observations outside of the interval between the lower limit and the upper limit, then it can be termed as outliers in the dataset.

Precision is the ratio of the correctly identified positive classes to the sum of the predicted positive classes. The predicted positive classes are the ones which are predicted positive irrespective of the actual value being positive or negative, that is, true positive and false positive classes. This ratio provides information that out of all the positive classes we have predicted correctly, how many are actually positive. 

Precision= TP/TP+FP

Recall is the ratio of the correctly identified positive classes to the sum of the actual positive classes. The actual positive classes can be predicted as positive or negative that is, true positive and false negative. This ratio provides the information that out of all the positive classes, how much we predicted correctly.

Recall= TP/TP+FN

The OLS assumptions for a linear regression are divided into five assumptions: 

• Linearity 

The regression assumes that the data is linear in nature. For higher degrees of variables, linear regression will not produce good predictions. 

• No endogeneity 

The issue of endogeneity arises when we have a variable that is related to the target and also the predictors but not included in the model. Therefore, endogeneity is a situation in which a predictor in a linear regression model is correlated to the error term. We call such predictors as endogenous variables. 

• Normality and homoscedasticity 

This assumes that the error term is normally distributed, and the expected value of error is 0, meaning that we expect to have no error on average. Homoscedasticity assumes that the variance is constant for the error term. 

• No autocorrelation 

This assumes that the covariance between two error terms is not zero. 

• No multicollinearity 

When two or more variables in our regression are strongly correlated, this situation is referred to as multicollinearity. The OLS assumptions assume that there are no strongly correlated variables in our analysis. 

One can standardise any distribution. The process of standardisation involves transforming the variables to one with a mean of zero and a standard deviation of one. 

Standardization is also possible for normal distributions. The result is known as a standard normal distribution. A standard normal distribution is represented by the letter Z. The Z-score is referred to as the standardised variable. The formula for standardising variables is defined by the Z-score. We first determine a variable's mean and standard deviation. The mean is then subtracted from each observed value of the variable, and then divide by the standard deviation.  

Exploratory data analysis is a method of data analysis in which the properties and qualities of the data are examined without making any attempt to fit the data to a specific model. This method's guiding principle is to analyse the data before using a particular model. We use numerical and graphical processes in exploratory data analysis to better comprehend the data. The emphasis on visual data representation has been a key component in the development of exploratory data analysis.

Skewed data refers to the data that contains outliers. Outliers are known to have a negative influence on the model’s predictions and thus needs to be eliminated. However, it is not always advised to remove the outliers, so we need to handle them through certain transformations. The common transformations applied on the data are log transformation, square root transformation, and box-cox transformation. 

• Log transformation 

The logarithmic transformation is one of the most helpful and popular transformations. In fact, it might be a good idea to use the dependent variable's logarithm as a replacement before doing a linear regression. A similar operation would stabilise the target variable's variance and bring the transformed variable's distribution closer to normal. 

• Square root transformation 

If there are any outlier values that are exceptionally large, you might consider using the square root transformation. The transformation can help scaling them down to a much lower value in comparison. A limitation of this transformation is that the square root of a negative number is not a real number.  

• Box-Cox transformation 

Box-cox transformation is yet another transformation method that can help to transform skewed data into normal. It has a controlling parameter lambda which ranges between -5 to 5. Initially, it was only used in presence of positive values, but modifications have been made to the transformation to take care of the negative values as well. 

The likelihood of an event provided that another event is known to have occurred is known as conditional probability. The expression "probability of A given that B has already occurred" refers to the conditional probability, which is written as P(A|B). When determining the likelihood of an event when a piece of information is already known, either entirely or partially, conditional probabilities are helpful. Examples include, determining the probability of getting the number 5 on the second throw provided that we have already got the number 6 on the first throw, drawing a red ball provided that the first two balls drawn are blue and green.

In this case, we have total 15 balls. The sample space will be the number of ways in which we can draw three balls from the bag of 15 balls. This can be done in 15C3 ways which is equal to 455. Now, the event of getting three red balls implies that we are trying to draw three red balls out of the 5 red balls present in the bag. We can draw three red balls among 5 red balls in 5C3 ways, equal to 10. The final result will be the number of possibilities of drawing three red balls divided by the total possibilities, that is, 10/455 or 2/91.

The strength of the linear relationship between two random variables is gauged by the simple correlation coefficient, commonly known as Pearson correlation. The range [1; 1] is where values of the correlation coefficient can exist. The two extreme values of this interval represent a positively skewed and negatively skewed perfectly linear relationship between the variables. The number zero indicates that there is no linear relationship. A non-parametric measuring correlation is the Spearman rank correlation coefficient. It is also used to establish the relationship that exists between two pieces of data. The Spearman rank correlation also works with monotonic or non-linear functions, unlike the Pearson coefficient, which requires a linear relationship between the two variables.

The F1 score is calculated as the harmonic mean of the precision and recall values. The mean or simple average treats all values equally. On the other hand, the harmonic mean gives more weight to the low values. As a result, the classifiers will only get a higher F1 score if both recall and precision is high.

The Akaike information criterion (AIC), a refined method based on in-sample fit, is used to determine how likely it is for a model to estimate or predict future values. Another model selection criterion that assesses the trade-off between model fit and complexity is the Bayesian information criterion (BIC). We utilise either the AIC or the BIC, but not both concurrently and interchangeably, to compare models with one another. The model that has the lowest AIC or BIC of all the models is a good model.

When two or more variables in our regression are strongly correlated, this situation is referred to as multicollinearity. The effect of multicollinearity among our variables is measured by the variance inflation factor, or VIF score. It gauges how much a predicted regression coefficient’s variance rises in the presence of correlation. We aren’t performing well if the variance of our model rises. A general rule of thumb that is frequently applied in practise is that high multicollinearity is present if the VIF score is greater than 10.

Imbalance classes are skewed classes where a single value might make up a significant amount of the data set, also known as the majority class. Consider that we are utilising a dataset of credit card fraud. The percentage of fraud incidents in the overall amount of data can be as low as 1%. In this situation, our model would still be 99% accurate if it were to blindly forecast each case to be authoritative. In order to prevent this, we either under sample the non-fraudulent instances or oversample the fraudulent ones, allowing them to make up a sizable fraction of the population.

The bias is defined as the difference between the actual and the predicted value of a variable. The bias shows how far the estimate is off from the variable’s actual value. Bias represents the assumptions made by the model to make the target function easier to learn. The variance calculates the mean deviations of a variable's values from its mean. It is the amount that the target function will change if a different training data is used. Bias and variance are prediction errors of an algorithm. The bias-variance trade-off is the property of the algorithm which suggests that these two errors should be minimized to avoid the learning algorithm from overfitting and underfitting the training data. An ideal model will have low bias and low variance. A low variance and high bias suggest overfitting while a low bias and high variance will mean that we are overfitting the data. Decreasing bias leads to increase in the variance and vice versa. Therefore, we need to find a balance where both these errors are minimum.

A resampling technique called cross-validation uses several data subsets to evaluate and train a model across a number of iterations. It is typically applied in situations where the objective is prediction, and one wishes to evaluate how well a predictive model will function in real-world situations. Due to sampling variability between training and test set, our model gives better prediction on training data but fails to generalize on test data. This leads to low training error rate and high test error rate. When we split the dataset into training, validation and test set, we only use a subset of data. To overcome these issues, we can adopt various cross validation approaches, namely, K-fold cross validation, stratified k-fold cross validation, leave one out cross validation, stratified shuffle split, etc. 

• Leave One Out Cross Validation (LOOCV) 

A dataset with n observations is divided into n-1 observations as the training data and 1 observation as test data. The process is iterated for each data point. Therefore, the execution is expensive. Also, for an outlier in test data, variability in MSE is much higher. 

• K-Fold Cross Validation 

Randomly divides data in k-groups or folds of equal sizes. The first fold is kept for testing and the model is trained on k-1 folds. The process is repeated k-times and each time different fold or a different group of data points are used for validation. Typically, k in k-fold is 5 or 10. LOOCV is a variant of k-fold where k = n. Though it is less computationally expensive than the LOOCV but still it is. 

• Stratified K-fold Cross Validation 

Each fold in the dataset has at least m instances of each class. This approach ensures that one class of data is not over-represented especially when the target variable is unbalanced.

Parametric models make assumptions of the underlying distribution and consists of fixed learning parameters. These models often have high bias and low variance which makes them prone to underfitting. Linear regression is an example of a parametric model since it assumes the underlying data distribution to be linear.

Non-parametric models do not make any assumptions for the data. Instead, they are free to learn but controlled by some hyperparameters. These models often have low bias and high variance which makes them prone to overfitting. Decision trees is an example of non-parametric models.

Consider a coin that has a slightly biased chance of landing on its head (51%), and a slightly biased chance of landing on its tail (49%). If you toss it 100 times, you will get 51 heads and 49 tails, or mostly heads. But for a small number of trials, this might not be the case. We can see that the likelihood of receiving a head will be 51% with a disproportionately large number of trials. With each further coin toss, the likelihood rises. According to Bernoulli's theorem, which is a condensed version of the law of large numbers, as the number of Bernoulli trials approaches infinity, the relative frequency of success in a series of trials approaches the chance of success.

Autocorrelation is a measure of the correlation of a particular time series with the same time series delayed by k lags. It is calculated by dividing the covariance between the current value and the lagged value by the standard deviation of the two observations, which are separated by k lags in a time series. We may determine the autocorrelation function by calculating the autocorrelation for all values of k. The autocorrelation function for a time series that doesn't vary over time falls off exponentially to zero.

If a timeseries' mean, variance, and covariance remain constant across time or do not change with passing intervals, the data is said to be stationary. The Dickey-Fuller test is a typical stationarity test. If the data is non-stationary, the initial step is to apply X-order of differencing to the data. We can keep comparing until we reach at stationarity. Each stage of the differencing process, however, results in the loss of one row of information. We can also make a seasonal difference for data that is showing a seasonal trend. If we had monthly data with yearly seasonality, for instance, we could differ by a time unit of 12, rather than just 1.

Bayes' theorem provides a formula for the likelihood that an occurrence is the direct outcome of a given condition if we take into account the set of conditions that an event occurs. So, it is possible to think of Bayes' theorem as a formula for the conditional probability of an occurrence. For instance, let us consider there are 10 bags containing different coloured marbles. Bayes’ theorem helps to determine the probability of drawing the marble from a particular bag, given the condition that the marble is red in colour. If A is the event of drawing the marble from a particular bag and B is the event of drawing a red marble, then the formula for Bayes’ theorem is given by –

P(A|B)=P(B|A)∙P(A)/P(B)

Where P(A|B) is the probability of event A to occur given the condition that event B has already occurred.

The following procedures are typically used when evaluating hypotheses about a sample: 

• Formulate the null hypothesis (H0) and alternate hypothesis (H1). 
• Determine the significance level, alpha for the hypothesis test. 
• Check for one-tailed or two-tailed test based on the null hypothesis. 
• Compute the critical value for the null hypothesis. 
• Determine the acceptance and rejection regions in accordance to the critical value. 
• Compare the sample statistics with the significance level, if it falls in the acceptance region then accept the null hypothesis, else reject it. 

We might also come across data that aren't normally distributed. There is a need for alternative techniques since they are more suited to examining the discrepancies between expected and observed frequencies. A statistical test that is non-parametric is one in which no assumptions are made on particular parameter values. One of the easiest and most used non-parametric tests in statistical research is the Chi-square test. The chi-square distribution is a continuous probability distribution that is distributed normally. The chi-square distribution moves closer to a normal distribution as ‘n’, the number of degrees of freedom, approaches towards infinity. The goodness of fit test, which compares observed frequencies and hypothetical frequencies of particular classes, is one of many methods for assessing hypotheses that uses the chi-square distribution. Additionally, it is used to assess the independence of two variables and to compare the observed variance and the hypothetical variance of samples with normally distributed data.

Clustering is the division of a data collection into subsets or clusters so that, according to an established distance metric, the degree of association is strong between members of the same cluster and weak between members of other clusters. There are several ways to carry out cluster analysis like partitional clustering, hierarchical clustering, etc. We must establish a distance between the objects that need to be categorised in order to do cluster analysis on a set of ‘n’ objects. It is expected that the collection of objects contains some sort of organisation. The distance between two classes in the single link technique is determined by the Euclidean distance between the two closest items in the distance table. In the complete linkage method, the distance between two classes is given by the Euclidean distance between the two elements furthest away.

The analysis of variance, also known as ANOVA, is an effective statistical method for significance testing. Only the significance of the difference between two sample means may be tested using the t-distribution-based test of significance. The hypothesis that all the samples are taken from the same population, i.e., they have the same mean, needs to be tested using a different approach when we have three or more samples to take into account at once. Therefore, the basic goal of the analysis of variance is to examine the homogeneity of three or more means.

Time-series data is data that is collected at different points in time. Autocorrelation, Seasonality, Stationarity are the three main components of a time series. 

• Autocorrelation 

Autocorrelation refers to the similarity between observations as a function of the time lag between them. We can find the value of a point by finding the period in the plot. 

• Seasonality 

Seasonality refers to periodic fluctuations. Period can give the length of the season. For instance, the amount of electricity consumed varies greatly from summer to winter, and online sales peak around Diwali before dipping again. 

• Stationarity  

Stationary means that statistical properties do not change over time, that is, constant mean and variance, and covariance is independent of time. For example, stock prices are not a stationary process. For modelling, we would prefer to have a stationary time series. However, there are other transformations we can apply to make them stationary. 

The Three Sigma Rule, sometimes known as the empirical rule, states that for a Normal Distribution, 68% of the data will be within one standard error of the mean. The data will be within two standard deviations of the mean in 95% of the cases. The probability that the data will be within three standard deviations of the mean is approximately 99.7%. For example, an RMSE equal to 50 means that about 68% of the system’s predictions fall within 50 of the actual value, about 95% of the predictions fall within 100 of the actual value, and about 99.7% of the predictions fall within 150 of the actual value.

The statistical field of survival analysis examines how long it is likely to be before an event occurs. Survival analysis is also known as time-to-event analysis. In a survival study, the length of time it takes for an event to occur is a key factor. The event we are usually interested in is death or failure. For instance, determining when a person will pass away following a diagnosis of a sickness or the failure of an appliance.

The survival function is estimated by Kaplan Meier curves. The survival function is graphically represented by the Kaplan-Meier curve. It displays the likelihood that a subject will live until time t. Plotting the survival function against time leads to the formation of the curve.

Operations research is an area of applied mathematics that makes use of scientific techniques to offer a foundation for making decisions. In order to discover the optimum approach to accomplish a task, it is frequently applied to complicated issues involving the organisation of personnel and equipment. Simulation, optimization, linear programming, nonlinear mathematical programming, game theory, and other techniques are all included in operations research approaches.

When two variables are correlated while accounting for a third variable or more, it is said to be partial correlation. A third variable Z acts as a measure of the direct relationship between variables X and Y in a measure of partial correlation between them, but it does not take into consideration the effects of their linear relationships with Z.