## Some Essential Numerical Summaries in Statistics for Data Science (Theory Python and R)Data science is a multidisciplinary subject that combines data, computer science, and domain information to extract insights, patterns and trends from data which could be useful in several domains which relies on historical data to capture trends and using the entire concept to predict future data. It involves a systematic technique of accumulating, cleaning, reading, and decoding vast quantities of information to aid in decision-making, anticipate trends, and discover hidden patterns. With the exponential boom of data generated through digital structures, sensors, and numerous technologies, the function of data science know-how has emerged as critical throughout industries including healthcare, finance, advertising, and extra. At its centre, data science know-how targets to convert raw data into actionable information. This transformation involves several key steps: data series, information preprocessing, exploratory data evaluation (EDA), modelling, and deployment. Each step requires a mix of technical talents, domain understanding, and analytical wondering. ## Mathematics for Data ScienceMathematics is the foundation upon which data technology is constructed. It provides the necessary equipment and techniques for studying information, constructing fashions, and making knowledgeable decisions. The most applicable branches of arithmetic in data science know-how include: **Statistics:**This department deals with the gathering, analysis, interpretation, and presentation of data. Concepts which include probability, distributions, speculation checking out, and regression evaluation are essential to information and deriving insights from data.**Linear Algebra:**Many data science algorithms, particularly in machine mastering, rely upon linear algebra. Operations with matrices and vectors, eigenvalues, and eigenvectors are critical for duties along with dimensionality discount (e.G., PCA) and know-how the workings of algorithms like Singular Value Decomposition (SVD).**Calculus:**Calculus, especially differential calculus, is vital for optimising algorithms in machine getting to know. Concepts like gradients and derivatives are utilised in gradient descent, that's a method to minimise blunders functions in schooling fashions.**Probability Theory:**This field underpins many statistical techniques and device learning algorithms. Understanding probability distributions, Bayes' theorem, and Markov techniques is important for making predictions and assessing the uncertainty of consequences.
## Numerical Summaries in Data ScienceNumerical summaries are statistical measures that provide a quick and concise description of a dataset's characteristics. These summaries are crucial in exploratory data evaluation (EDA), assisting information scientists to understand the important tendency, dispersion, and form of the statistics distribution. Key numerical summaries encompass imply, median, mode, variance, popular deviation, range, interquartile variety (IQR), percentiles, skewness, and kurtosis. ## Importance of Numerical SummariesNumerical summaries are important equipment in data science, presenting vital insights into the traits of a dataset. These summaries help information scientists to understand the statistics at a look, facilitating more knowledgeable decision-making.
**Simplifying Complex Data:**Large datasets may be overwhelming and tough to interpret. Numerical summaries distil these complex datasets into less difficult, more attainable metrics. By summarising key factors of the data, which includes primary tendency, variability, and distribution shape, data scientists can quickly grasp the general patterns and developments without getting bogged down with the aid of man or woman information points.**Identifying Data Characteristics:**Numerical summaries provide a snapshot of the data's key traits, inclusive of:**Central Tendency:**Measures just like the mean, median, and mode indicate the everyday or crucial value within the dataset. This allows information about what constitutes a "regular" or "expected" value.**Dispersion:**Measures like variance, standard deviation, range, and interquartile variety (IQR) screen how unfold out the information points are.
**Data Cleaning and Preprocessing:**Numerical summaries are crucial inside the data cleansing and preprocessing levels. They help identify:**Outliers:**Extreme values that deviate significantly from other observations. Outliers can distort evaluation and modelling if no longer well addressed.**Missing Values:**Summaries inclusive of the count and percent of missing values can highlight gaps within the data, guiding selections on imputation or exclusion.**Data Distribution:**Understanding the distribution of data is crucial for choosing the right statistical strategies and fashions. For example, typically distributed data may be suitable for parametric assessments, at the same time as non-everyday information would possibly require non-parametric strategies.
**Comparing Different Datasets:**Numerical summaries permit for quick evaluation between extraordinary datasets or subsets of information. For instance, evaluating the imply and variance of income information across exclusive regions can monitor local performance variations. This comparative analysis is crucial for figuring out developments, making predictions, and formulating strategies.**Hypothesis Testing and Inferential Statistics:**Many inferential statistical techniques rely on numerical summaries. For example, t-tests and ANOVAs compare means to determine if differences between groups are statistically significant. Summaries like mean and variance are foundational for these tests, making them essential for hypothesis testing.**Model Building and Evaluation:**Numerical summaries play a crucial role in the model-building process:**Feature Selection:**Understanding the variance and distribution of capabilities facilitates in choosing the most informative variables for modelling.**Normalisation and Standardization:**Measures which include imply and fashionable deviation are used to normalise information, ensuring that features make contributions similarly to the model.**Model Evaluation:**Summaries like the mean absolute error (MAE) and root mean squared error (RMSE) are used to evaluate version performance, supplying metrics that quantify how properly a model predicts effects.
**Communicating Results:**Clear and concise numerical summaries are crucial for speaking findings to stakeholders. Whether in reports, presentations, or dashboards, these summaries offer a reachable manner for non-technical audiences to understand the statistics. They facilitate data-driven decision-making by means of highlighting key insights in a without difficulty interpretable form.**Practical Examples:****Mean and Median:**In a commercial enterprise placing, the imply and median sales figures can help understand traditional income performance, even as additionally highlighting capability discrepancies because of outliers.**Variance and Standard Deviation:**In finance, the variance and trendy deviation of asset returns are vital for assessing chance. Higher variability shows better danger, that's important for investment decisions.**Skewness and Kurtosis:**In fine management, skewness and kurtosis help in know-how the distribution of product defects, guiding enhancements in production strategies.**Interquartile Range (IQR):**In healthcare, the IQR of patient recuperation instances can provide insights into the consistency of remedy effectiveness, supporting to pick out best practices.
## Understanding Mean## Definition:The imply, frequently known as the common of information, is a degree of central tendency that is calculated by taking the sum of all data points divided with the aid of the variety of data points within the statistics sample. Mathematically, it's expressed as: ## Importance of Mean:**Central Tendency:**The suggestion offers a concept of the principal cost of the dataset, offering a short photo of the dataset's well-known behaviour.**Comparison:**It permits for easy assessment between distinct datasets or special corporations in the same dataset. For example, comparing the suggested income across special areas can offer insights into local monetary disparities.**Statistical Analysis:**The mean is foundational in various statistical methods and hypothesis testing. It is utilised in calculating other statistical measures including variance and trendy deviation, which might be crucial for information records spread.**Simplicity:**The mean is simple to compute and understand, making it widely used in basic data analysis. Its simplicity makes it a starting point for more complex statistical analysis and machine learning models.
## Calculation Methods:The mean is calculated using the formula mentioned earlier. **Sum:**Add up all the values in the dataset.**Divide by the number of data points:**Take the total sum and divide it by the count of values in the dataset.
## Example:For a dataset [1, 2, 3, 4, 5]: - Sum = 1 + 2 + 3 + 4 + 5 = 15
- Number of data points (n) = 5
- Mean = 15 / 5 = 3
## Implementation of Mean in Python:Here's how to calculate the mean in Python using the NumPy library:
Mean: 3.0 ## Implementation of Mean in R:In R, the mean can be calculated using the mean function:
[1] "Mean: 3" ## Understanding Median## Definition:The median is every other degree of primary tendency that represents the centre value of a dataset whilst its miles are ordered in ascending or descending order. If the dataset has an abnormal variety of observations, the median is the centre number. If the dataset has an excellent wide variety of observations, the median is the common of the two middle numbers. ## Importance of Median:**Robustness to Outliers:**Unlike the imply, the median isn't always laid low with intense values or outliers, making it a higher degree of relevant tendency for skewed distributions.**Central Tendency:**It presents a clear indication of the middle of the dataset, imparting a higher sense of the data's critical value when the data isn't always symmetrically distributed.**Statistical Analysis:**The median is used in non-parametric data and is especially beneficial in descriptive data for skewed distributions. It is often used along with other measures to provide a comprehensive know-how of the data.**Decision Making:**In real-international programs, consisting of income distribution or property prices, the median offers an extra accurate mirrored image of the standard value. For example, the median profits is regularly used in place of the suggested earnings to represent the everyday income of a populace, as it isn't always skewed by means of very excessive or very low values.
## Calculation Methods:To calculate the median: - Sort the data in ascending order.
- Determine the middle value:
- If the number of observations (nnn) is odd, the median of the distribution is the middle value in the data.
- If the number of observations is even, the median value is the average of the two middle values in the data.
## Example:For a dataset [45, 67, 23, 89, 90]: - Sorted data: [23, 45, 67, 89, 90]
- Number of data points (n) = 5 (odd)
- Median = 67 (the middle value)
For a dataset [1, 2, 3, 4, 5, 6]: - Sorted data: [1, 2, 3, 4, 5, 6]
- Number of data points (n) = 6 (even)
- Median = (3 + 4) / 2 = 3.5
## Implementation of Median in Python:Here's how to calculate the median in Python using the NumPy library:
Median: 3.0 ## Implementation of Median in R:In R, the median can be calculated using the median function:
[1] "Median: 3" ## Understanding Mode## Definition:The mode is a measure of relevant tendency that represents the maximum regularly taking place value in a dataset. Unlike the mean and median, which are measures of the central area, the mode makes a speciality of the frequency of values. A dataset will have one mode (unimodal), two modes (bimodal), or extra (multimodal). In some cases, especially with continuous data, there might be no mode at all if no wide variety repeats. ## Importance of Mode:**Categorical Data:**The mode is particularly beneficial for categorical information wherein we want to recognize which is the maximum common class. For example, in a survey of favourite colours, the mode would pick out the shade maximum human's pick.**Understanding Distribution:**The mode gives insights into the distribution of the information. For instance, in a dataset of shoe sizes bought, the mode tells us the most commonplace shoe size offered.**Decision Making:**In commercial enterprise and economics, understanding the mode can help in stock management, product design, and advertising strategies. For example, knowing the most common size of a product helps in stock management.**Robustness:**The mode is not affected by outliers or extreme values, making it a stable measure of central tendency in certain scenarios.
## Calculation Methods:To calculate the mode: **Tally the frequencies:**Count the number of occurrences of each value in the dataset.**Identify the highest frequency:**The value(s) with the highest count is the mode.
## Example:For a dataset [1, 2, 2, 3, 4]: - Tally: 1 occurs once, 2 occurs twice, 3 occurs once, 4 occurs once.
- Highest frequency: 2 (it occurs twice).
- Mode = 2
## Implementation of Mode in Python:Here's how to calculate the mode in Python using the scipy.stats module:
Mode: 2 Frequency: 2 Alternatively, using pandas:
Mode: 2 ## Implementation of Mode in R:In R, the mode can be calculated using custom functions since base R does not have a built-in mode function:
[1] "Mode: 2" ## Understanding Standard Deviation## Definition:Standard deviation is a measure of the amount of variation or dispersion in a set of values. It quantifies how much the individual data points in a dataset differ from the dataset's mean. Mathematically, the standard deviation is the square root of the variance. For a dataset with n observations, the formula for the standard deviation (?) is as follows: Where: - x
_{i}represents each data point and - μ is the mean of the dataset. For a sample from a population, the formula adjusts to use n -1
- n -1 in the denominator instead of n to provide an unbiased estimate.
## Importance of Standard Deviation:**Quantifying Variability:**Standard deviation provides a measure of the dispersion of statistics points across the suggest, indicating how spread out the values are. A small trendy deviation indicates that the values are close to the suggested, while a large fashionable deviation indicates a huge variety of values.**Risk Assessment:**In finance, fashionable deviation is used to measure the volatility of an investment. A better general deviation indicates a higher danger, as the investment's returns are extra spread out.**Quality Control:**In production and excellent manipulation, preferred deviation enables in tracking processes to ensure consistency and identifying versions that can require corrective movements.**Comparing Distributions:**Standard deviation permits for the comparison of variability between extraordinary datasets or distributions. It enables information whether one dataset is a greater variable than every other.**Statistical Inference:**Standard deviation is utilised in various statistical analyses, such as hypothesis checking out and self-belief durations. It allows in making inferences about populations from sample statistics.**Normalisation and Standardization:**Standard deviation is utilised in data preprocessing steps consisting of normalisation and standardisation, making it critical for system learning and information analysis.
## Calculation Methods:The standard deviation is calculated through the following steps: **Calculate the mean (μ):**Sum all data points and divide by the number of points.**Calculate each point's deviation from the mean:**Subtract the mean from each data point.**Square each deviation:**This eliminates negative values and emphasises larger deviations.**Sum all squared deviations:**Add up all the squared deviations.**Divide by the number of data points (or n - 1 for a sample):**This gives the variance.**Take the square root of the variance:**This is the standard deviation.
## Implementation of Standard Deviation in Python:Here's how to calculate the standard deviation in Python using the NumPy library:
Standard Deviation: 1.5811388300841898 Alternatively, using pandas:
Standard Deviation: 1.5811388300841898 ## Implementation of Standard Deviation in R:In R, the standard deviation can be calculated using the sd function:
[1] "Standard Deviation: 1.58113883008419" ## Understanding Variance## Definition:Variance is a statistical measure that quantifies the dispersion of data points in a dataset relative to the mean. It indicates how much the values in the dataset differ from the average value. Mathematically, variance is the average of the squared differences from the mean. For a dataset with n observations, the variance ? Where: - x
_{i}represents each data point, and - μ is the mean of the dataset.
- For a sample from a population, the formula adjusts to use n -1 in the denominator instead of n to provide an unbiased estimate.
## Importance of Variance:**Understanding Dispersion:**Variance provides a quantitative measure of the spread of data points. It helps in understanding how data points are distributed around the mean.**Risk Assessment:**In finance, variance is used to measure the volatility of an investment. A higher variance indicates higher risk, as returns are more spread out.**Quality Control:**In manufacturing, variance is used to monitor and control the quality of products. Lower variance indicates consistent product quality.**Statistical Inference:**Variance is a crucial component in various statistical analyses, including hypothesis testing and confidence intervals.**Comparison of Datasets:**Variance allows comparison of the variability between different datasets. It helps in identifying which dataset is more consistent.**Data Analysis:**Variance is used in data preprocessing steps such as normalisation, where it helps in scaling features to have the same level of variance.
## Calculation Methods:To calculate the variance, follow these steps: **Calculate the mean:**Sum all data points and divide by the number of points.**Calculate each point's deviation from the mean:**Subtract the mean from each data point.**Square each deviation:**This eliminates negative values and emphasises larger deviations.**Sum all squared deviations:**Add up all the squared deviations.**Divide by the number of data points:**This gives the variance.
## Implementation of Variance in Python:Here's how to calculate the variance in Python using the NumPy library:
Variance: 2.5 Alternatively, using pandas:
Variance: 2.5 ## Implementation of Variance in R:In R, the variance can be calculated using the var function:
[1] "Variance: 2.5" ## Understanding Range## Definition:The range is a measure of statistical dispersion that represents the difference between the maximum and minimum values in a dataset. It provides a simple way to understand the spread or variability of the data. The formula for calculating the range is: Range = Maximum_Value - Minimum_Value For example, in a dataset [3, 7, 8, 2, 5], the range is 8 - 2 = 6. ## Importance of Range:**Understanding Data Spread:**The range gives a quick sense of the spread of the data. It tells us how far apart the extreme values are, providing a basic understanding of data variability.**Initial Data Analysis:**Range is often used in exploratory data analysis to get a preliminary idea about the dispersion of the dataset. It can highlight the presence of outliers.**Comparison Between Datasets:**When comparing two or more datasets, the range can provide insights into which dataset has more variability.**Identifying Outliers:**A large range might indicate the presence of outliers or extreme values in the dataset.**Basis for More Complex Measures:**While the range itself is a simple measure, it serves as a foundation for understanding more complex measures of variability like variance and standard deviation.
However, the range has limitations. It only considers the extreme values and ignores the distribution of the data between them. Therefore, it is often used in conjunction with other statistical measures to provide a more comprehensive analysis. ## Calculation Methods:To calculate the range: **Identify the Maximum Value:**Find the highest value in the dataset.**Identify the Minimum Value:**Find the lowest value in the dataset.**Subtract the Minimum from the Maximum:**The result is the range.
## Example:For the dataset [10, 15, 20, 2, 8]: - Maximum value = 20
- Minimum value = 2
- Range = 20 - 2 = 18
## Implementation of Range in Python:Here's how to calculate the range in Python using basic Python functions and the NumPy library:
Range: 18
Range: 18 ## Implementation of Range in R:In R, the range can be calculated using basic functions and the diff function:
[1] "Range: 18"
[1] "Range: 18" ## Understanding the Interquartile Range## Definition:The Interquartile Range (IQR) is a measure of statistical dispersion, which indicates the spread of the middle 50% of a dataset. It is calculated as the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 - Q10 Quartiles divide a ranked dataset into four equal parts. The first quartile (Q1) is the median of the lower half of the data (25th percentile), and the third quartile (Q3) is the median of the upper half of the data (75th percentile). The second quartile (Q2) is the median of the entire dataset. ## Importance of Interquartile Range:**Robust Measure of Dispersion:**Unlike the range, which only considers the extreme values, the IQR focuses on the central portion of the data, providing a more robust measure of variability that is less sensitive to outliers.**Identification of Outliers:**The IQR is used to identify outliers. Values that fall below**Q1 - 1.5 × IQR**or above**Q3 +****5 × IQR**are typically considered outliers.**Comparison of Distributions:**The IQR allows for comparison of the spread of different datasets. It helps in understanding how the middle 50% of the data varies across different groups.**Data Summarization:**By summarising the spread of the middle 50% of the data, the IQR provides a clear picture of the central tendency and dispersion without being affected by extreme values.**Use in Box Plots:**The IQR is a key component in creating box plots, which are graphical representations of data distributions. Box plots visually show the median, quartiles, and potential outliers.
## Calculation Methods:To calculate the IQR: **Arrange Data:**Sort the dataset in ascending order.**Find Quartiles:**- Q1: The median of the lower half of the data.
- Q3: The median of the upper half of the data.
**Calculate IQR:**Subtract Q1 from Q3.
## Example:For the dataset [7, 15, 36, 39, 40, 41, 42, 43, 47, 49]: **Arrange data:**[7, 15, 36, 39, 40, 41, 42, 43, 47, 49]**Find Q1 (25th percentile) and Q3 (75th percentile):**- Q1 = 37.5
- Q3 = 44.5
**Calculate IQR:**IQR = 44.5 - 37.5 = 7
## Implementation of Interquartile Range in Python:Here's how to calculate the IQR in Python using the NumPy and SciPy libraries:
Interquartile Range (IQR): 7.0
Interquartile Range (IQR): 7.0 ## Implementation of Interquartile Range in R:In R, the IQR can be calculated using the IQR function:
[1] "Interquartile Range (IQR): 7" ## Understanding Percentiles and Quartiles## Definitions:
A percentile is a measure used in statistics that indicates the value below which a given percentage of observations in a group of observations falls. For example, the 20th percentile is the value below which 20% of the observations may be found. Percentiles divide a dataset into 100 equal parts.
Quartiles are a type of quantile, which divide a dataset into four equal parts. The three quartiles are: **First Quartile (Q1):**The 25th percentile, below which 25% of the data falls.**Second Quartile (Q2 or Median):**The 50th percentile, below which 50% of the data falls.**Third Quartile (Q3):**The 75th percentile, below which 75% of the data falls.
The interquartile range (IQR) is the range between the first and third quartiles and is a measure of statistical dispersion. ## Importance of Percentiles and Quartiles:**Data Distribution:**Percentiles and quartiles provide insights into the distribution of data. They help in understanding how data points are spread out and where the majority of data points lie.**Outlier Detection:**Quartiles, particularly the IQR, are useful in detecting outliers. Data points that lie below Q1-5×IQRQ1 - 1.5 \times IQRQ1-1.5×IQR or above Q3+1.5×IQRQ3 + 1.5 \times IQRQ3+1.5×IQR are often considered outliers.**Comparative Analysis:**Percentiles are widely used in comparative analysis, such as comparing scores in standardised tests. They indicate the relative standing of a value within a dataset.**Summarising Data:**Quartiles summarise data by dividing it into four parts, making it easier to understand the spread and central tendency of the data.**Non-Parametric Statistics:**Percentiles and quartiles are non-parametric statistics, meaning they do not assume a specific distribution. This makes them useful for analysing data that does not follow normal distribution.
## Calculation Methods:
- Sort the data in ascending order.
- Use the formula
**P = (n + 1) × p / 100**, where n is the number of observations and p is the desired percentile. - Find the value at the P-th position in the sorted list.
- Sort the data in ascending order.
- Calculate Q1, Q2 (median), and Q3 using the 25th, 50th, and 75th percentiles, respectively.
## Example:For the dataset [7, 15, 36, 39, 40, 41, 42, 43, 47, 49]: - Q1 (25th percentile) = 36
- Q2 (50th percentile or median) = 40.5
- Q3 (75th percentile) = 43
## Implementation of Percentiles and Quartiles in Python:
25th Percentile (Q1): 36.0 50th Percentile (Q2): 40.5 75th Percentile (Q3): 43.0 ## Implementation of Percentiles and Quartiles in R:
[1] "25th Percentile (Q1): 36" [1] "50th Percentile (Q2): 40.5" [1] "75th Percentile (Q3): 43" ## Understanding Skewness## Definition:Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It quantifies how much a distribution deviates from a normal distribution, which is symmetrical. A distribution can be: **Positively Skewed (Right Skewed):**The right tail (higher values) is longer or fatter than the left tail (lower values). This indicates that the bulk of the values lie to the left of the mean.**Negatively Skewed (Left Skewed):**The left tail (lower values) is longer or fatter than the right tail (higher values). This indicates that the bulk of the values lie to the right of the mean.**Symmetrical:**The values are evenly distributed on both sides of the mean, indicating no skewness.
Mathematically, skewness can be calculated using the formula: Where: **n**is the number of observations,**x**is each individual observation,_{i}**x**is the mean, and**s**is the standard deviation.
## Importance of Skewness:**Understanding Data Distribution:**Skewness helps in understanding the distribution of data. It indicates whether the data is symmetric or if it leans more towards one side of the mean.**Identifying Outliers:**High skewness indicates the presence of outliers. Positively skewed data often have outliers on the higher end, while negatively skewed data have outliers on the lower end.**Data Transformation:**Knowing the skewness helps in deciding if data transformation is necessary. For instance, log transformation can be used to reduce positive skewness.**Model Selection:**Certain statistical models assume normality in the data. Understanding skewness helps in choosing the appropriate model or in applying transformations to meet model assumptions.**Financial Analysis:**In finance, skewness is used to assess the risk of investment returns. Positively skewed return distributions imply that there is a probability of extreme positive returns, while negatively skewed distributions imply a probability of extreme negative returns.
## Calculation Methods:To calculate skewness, follow these steps: - Calculate the mean.
- Calculate the standard deviation.
- Calculate the skewness using the skewness formula.
## Implementation of Skewness in Python:Here's how to calculate skewness in Python using the scipy.stats library:
Skewness: 0.531 Alternatively, using pandas:
Skewness: 0.531 ## Implementation of Skewness in R:In R, the skewness can be calculated using the e1071 package:
[1] "Skewness: 0.531" ## Understanding Kurtosis## Definition:Kurtosis is a statistical measure that describes the shape of a distribution's tails in relation to its overall shape. Specifically, it quantifies whether the data are heavy-tailed or light-tailed compared to a normal distribution. There are three types of kurtoses: **Mesokurtic:**Distributions with kurtosis similar to a normal distribution. Kurtosis value is approximately zero.**Leptokurtic:**Distributions with heavier tails and a sharper peak than a normal distribution. Kurtosis value is greater than zero.**Platykurtic:**Distributions with lighter tails and a flatter peak than a normal distribution. Kurtosis value is less than zero.
Mathematically, kurtosis is calculated using the formula: Where: **n**is the number of observations,**x**is each individual observation,_{i}**x**is the mean, and**s**is the standard deviation.
## Importance of Kurtosis:**Understanding Tail Risk:**Kurtosis helps in understanding the tail risk of a distribution. High kurtosis means there is a higher chance of extreme values (outliers).**Financial Risk Management:**In finance, kurtosis is used to assess the risk of investment returns. Leptokurtic distributions indicate higher risk due to potential extreme returns.**Data Analysis:**Kurtosis provides insights into the shape and nature of data distribution, which is essential for choosing appropriate statistical models.**Normality Testing:**Kurtosis, along with skewness, is used to test the normality of data. Data with high kurtosis may not follow a normal distribution, affecting the application of parametric tests.**Quality Control:**In manufacturing and quality control, kurtosis can indicate the likelihood of defects or deviations from the standard.
## Calculation Methods:To calculate kurtosis, follow these steps: - Calculate the mean.
- Calculate the standard deviation.
- Calculate each observation's deviation from the mean and raise it to the fourth power.
- Sum these values and apply the kurtosis formula.
## Implementation of Kurtosis in Python:Here's how to calculate kurtosis in Python using the scipy.stats library:
Kurtosis: -1.2685714285714287 Alternatively, using pandas:
Kurtosis: -1.2685714285714287 ## Implementation of Kurtosis in R:In R, the kurtosis can be calculated using the e1071 package:
[1] "Kurtosis: -1.26857142857143" Next Topic# |