Basic Descriptive Statistics
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This is a presentation for my Env. Chem students regarding the basic statistics they will be using in class throughout the year.
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There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
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In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models. Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression has many practical uses. Most applications fall into one of the following two broad categories:
If the goal is error reduction in prediction or forecasting, linear regression can be used to fit a predictive model to an observed data set of values of the response and explanatory variables. After developing such a model, if additional values of the explanatory variables are collected without an accompanying response value, the fitted model can be used to make a prediction of the response.
If the goal is to explain variation in the response variable that can be attributed to variation in the explanatory variables, linear regression analysis can be applied to quantify the strength of the relationship between the response and the explanatory variables, and in particular to determine whether some explanatory variables may have no linear relationship with the response at all, or to identify which subsets of explanatory variables may contain redundant information about the response.
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There are different distributions namely Normal, Skewed, and Binomial etc.
Objectives:
Normal distribution its properties its use in biostatistics
Transformation to standard normal distribution
Calculation of probabilities from standard normal distribution using Z table.
Normal distribution:
- Certain data, when graphed as a histogram (data on the horizontal axis, frequency on the vertical axis), creates a bell-shaped curve known as a normal curve, or normal distribution.
- Two parameters define the normal distribution, the mean (µ) and the standard deviation (σ).
Properties of the Normal Distribution:
Normal distributions are symmetrical with a single central peak at the mean (average) of the data.
The shape of the curve is described as bell-shaped with the graph falling off evenly on either side of the mean.
Fifty percent of the distribution lies to the left of the mean and fifty percent lies to the right of the mean.
-The mean, the median, and the mode fall in the same place. In a normal distribution the mean = the median = the mode.
- The spread of a normal distribution is controlled by the
standard deviation.
In all normal distributions the range ±3σ includes nearly
all cases (99%).
Uni modal:
One mode
Symmetrical:
Left and right halves are mirror images
Bell-shaped:
With maximum height at the mean, median, mode
Continuous:
There is a value of Y for every value of X
Asymptotic:
The farther the curve goes from the mean, the closer it gets to the X axis but it never touches it (or goes to 0).
The total area under a normal distribution curve is equal to 1.00, or 100%.
Using Normal distribution for finding probability:
While finding out the probability of any particular observation we find out the area under the curve which is covered by that particular observation. Which is always 0-1.
Transforming normal distribution to standard normal distribution:
Given the mean and standard deviation of a normal distribution the probability of occurrence can be worked out for any value.
But these would differ from one distribution to another because of differences in the numerical value of the means and standard deviations.
To get out of this problem it is necessary to find a common unit of measurement into which any score could be converted so that one table will do for all normal distributions.
This common unit is the standard normal distribution or Z
score and the table used for this is called Z table.
- A z score always reflects the number of standard deviations above or below the mean a particular score or value is.
where
X is a score from the original normal distribution,
μ is the mean of the original normal distribution, and
σ is the standard deviation of original normal distribution.
Steps for calculating probability using the Z-
score:
-Sketch a bell-shaped curve,
- Shade the area (which represents the probability)
-Use the Z-score formula to calculate Z-value(s)
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The mean is the average value found by dividing the sum of all values by the total number of values. The median is the middle value when data is arranged in order. For even datasets, the median is the average of the two middle values. The mode is the value that occurs most frequently in the dataset.
Lecture 3 & 4 Measure of Central Tendency.pdf
Lecture 3 & 4 Measure of Central Tendency.pdfkelashraisal This document provides an overview of measures of central tendency including average, mean, median, mode, and midrange. It defines each measure and provides examples of calculating them for both individual values and grouped data. The mean is the sum of all values divided by the total number of values. The median is the middle value of data in ascending order. The mode is the most frequent value. The midrange is the average of the minimum and maximum values. Formulas are given for calculating each measure for both individual data and grouped frequency distributions.
Statistical Methods
Statistical Methodsguest2137aa 1. Statistics is used to analyze data beyond what can be seen in maps and diagrams by using mathematical manipulation, which can reveal patterns that may otherwise go unnoticed.
2. It is important to justify any statistical techniques used and to ensure the data is appropriate for the technique. Students should ask what the technique can prove and if the data is in the right format before performing calculations.
3. Common methods for summarizing a large data set are the mean, median, and mode. The mean is the average, the median is the middle value, and the mode is the most frequent value. These give a single value for the data but do not show the variation around that value.
Statistical Methods
Statistical Methodsguest9fa52 1. Statistics is used to analyze data beyond what can be seen in maps and diagrams by using mathematical manipulation, which can reveal patterns that may otherwise go unnoticed.
2. It is important to justify any statistical techniques used and to only use techniques that are appropriate for the type of data.
3. Common methods for summarizing large data sets include calculating the mean, mode, and median. The mean is the average, the mode is the most frequent value, and the median is the middle value when the data is arranged from lowest to highest.
3.1-3.2 Measures of Central Tendency
3.1-3.2 Measures of Central Tendencymlong24 This document discusses measures of central tendency including the mean, median, and mode. It provides examples of calculating the mean from raw data sets and frequency distributions. The median and mode are defined as the middle value and most frequent value, respectively. Methods for calculating each from both types of data are shown. Other measures covered include the midrange and the effects of outliers. Shapes of distributions are discussed including positively and negatively skewed and symmetric. Practice problems are provided to reinforce the concepts.
Dr digs central tendency
Dr digs central tendencydrdig This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is the average and is calculated by summing all values and dividing by the total number of data points. The median is the middle value when data is arranged in order. The mode is the value that occurs most frequently in the data set. Examples are given to demonstrate calculating each measure. The document also discusses advantages and limitations of each central tendency measure.
A General Manger of Harley-Davidson has to decide on the size of a.docx
A General Manger of Harley-Davidson has to decide on the size of a.docxevonnehoggarth79783 A General Manger of Harley-Davidson has to decide on the size of a new facility. The GM has narrowed the choices to two: large facility or small facility. The company has collected information on the payoffs. It now has to decide which option is the best using probability analysis, the decision tree model, and expected monetary value.
Options:
Facility
Demand Options
Probability
Actions
Expected Payoffs
Large
Low Demand
0.4
Do Nothing
($10)
Low Demand
0.4
Reduce Prices
$50
High Demand
0.6
$70
Small
Low Demand
0.4
$40
High Demand
0.6
Do Nothing
$40
High Demand
0.6
Overtime
$50
High Demand
0.6
Expand
$55
Determination of chance probability and respective payoffs:
Build Small:
Low Demand
0.4($40)=$16
High Demand
0.6($55)=$33
Build Large:
Low Demand
0.4($50)=$20
High Demand
0.6($70)=$42
Determination of Expected Value of each alternative
Build Small: $16+$33=$49
Build Large: $20+$42=$62
Click here for the Statistical Terms review sheet.
Submit your conclusion in a Word document to the M4: Assignment 2 Dropbox byWednesday, November 18, 2015.
A General Manger of Harley
-
Davidson has to decide on the size of a new facility. The GM has narrowed
the choices to two: large facility or small facility. The company has collected information on the payoffs. It
now has to decide which option is the best u
sing probability analysis, the decision tree model, and
expected monetary value.
Options:
Facility
Demand
Options
Probability
Actions
Expected
Payoffs
Large
Low Demand
0.4
Do Nothing
($10)
Low Demand
0.4
Reduce Prices
$50
High Demand
0.6
$70
Small
Low Demand
0.4
$40
High Demand
0.6
Do Nothing
$40
High Demand
0.6
Overtime
$50
High Demand
0.6
Expand
$55
A General Manger of Harley-Davidson has to decide on the size of a new facility. The GM has narrowed
the choices to two: large facility or small facility. The company has collected information on the payoffs. It
now has to decide which option is the best using probability analysis, the decision tree model, and
expected monetary value.
Options:
Facility
Demand
Options
Probability Actions
Expected
Payoffs
Large Low Demand 0.4 Do Nothing ($10)
Low Demand 0.4 Reduce Prices $50
High Demand 0.6
$70
Small Low Demand 0.4
$40
High Demand 0.6 Do Nothing $40
High Demand 0.6 Overtime $50
High Demand 0.6 Expand $55
SAMPLING MEAN:
DEFINITION:
The term sampling mean is a statistical term used to describe the properties of statistical distributions. In statistical terms, the sample meanfrom a group of observations is an estimate of the population mean. Given a sample of size n, consider n independent random variables X1, X2... Xn, each corresponding to one randomly selected observation. Each of these variables has the distribution of the population, with mean and standard deviation. The sample mean is defined to be
WHAT IT IS USED FOR:
It is also used to measure central tendency of the numbers in a .
classroom 2.pptx
classroom 2.pptxdennissombilon1 The document defines and provides examples of calculating the mean, median, and mode of data sets. It discusses how the mean is the average, median is the middle value, and mode is the most frequent value. Examples are provided to demonstrate calculating each measure of central tendency for both ungrouped and grouped data. An assessment with multiple choice and short answer questions is also included to test understanding of these concepts.
BA 3 Statistics.ppt
BA 3 Statistics.pptNimeshGandhey The document discusses key concepts in statistics including:
- Statistics uses mathematical tools to describe and analyze data. It assumes populations follow a normal distribution.
- Parameters like the mean, standard deviation, and variance are used to characterize samples and populations.
- The standard deviation measures how spread out values are from the mean and defines the shape of the normal distribution. It indicates the precision of a data set.
analytical representation of data
analytical representation of dataUnsa Shakir This document discusses analytical representation of data through descriptive statistics. It begins by showing raw, unorganized data on movie genre ratings. It then demonstrates organizing this data into a frequency distribution table and bar graph to better analyze and describe the data. It also calculates averages for each movie genre. The document then discusses additional descriptive statistics measures like the mean, median, mode, and percentiles to further analyze data through measures of central tendency and dispersion.
1.0 Descriptive statistics.pdf
1.0 Descriptive statistics.pdfthaersyam This document provides an overview of descriptive statistics concepts and methods. It discusses numerical summaries of data like measures of central tendency (mean, median, mode) and variability (standard deviation, variance, range). It explains how to calculate and interpret these measures. Examples are provided to demonstrate calculating measures for sample data and interpreting what they say about the data distribution. Frequency distributions and histograms are also introduced as ways to visually summarize and understand the characteristics of data.
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Why manual and rule-based optimization approaches fall short in dynamic cloud environments
How machine learning predicts workload patterns to right-size resources before they're needed
Real-world implementation strategies that don't compromise reliability or performance
Featured case study: Learn how Palo Alto Networks implemented autonomous resource optimization to save $3.5M in cloud costs while maintaining strict performance SLAs across their global security infrastructure.
Bio:
Suresh Mathew is the CEO and Founder of Sedai, an autonomous cloud management platform. Previously, as Sr. MTS Architect at PayPal, he built an AI/ML platform that autonomously resolved performance and availability issues—executing over 2 million remediations annually and becoming the only system trusted to operate independently during peak holiday traffic.
GyrusAI - Broadcasting & Streaming Applications Driven by AI and ML
GyrusAI - Broadcasting & Streaming Applications Driven by AI and MLGyrus AI Gyrus AI: AI/ML for Broadcasting & Streaming
Gyrus is a Vision Al company developing Neural Network Accelerators and ready to deploy AI/ML Models for Video Processing and Video Analytics.
Our Solutions:
Intelligent Media Search
Semantic & contextual search for faster, smarter content discovery.
In-Scene Ad Placement
AI-powered ad insertion to maximize monetization and user experience.
Video Anonymization
Automatically masks sensitive content to ensure privacy compliance.
Vision Analytics
Real-time object detection and engagement tracking.
Why Gyrus AI?
We help media companies streamline operations, enhance media discovery, and stay competitive in the rapidly evolving broadcasting & streaming landscape.
🚀 Ready to Transform Your Media Workflow?
🔗 Visit Us: https://gyrus.ai/
📅 Book a Demo: https://gyrus.ai/contact
📝 Read More: https://gyrus.ai/blog/
🔗 Follow Us:
LinkedIn - https://www.linkedin.com/company/gyrusai/
Twitter/X - https://twitter.com/GyrusAI
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Challenges in Migrating Imperative Deep Learning Programs to Graph Execution:...
Challenges in Migrating Imperative Deep Learning Programs to Graph Execution:...Raffi Khatchadourian Efficiency is essential to support responsiveness w.r.t. ever-growing datasets, especially for Deep Learning (DL) systems. DL frameworks have traditionally embraced deferred execution-style DL code that supports symbolic, graph-based Deep Neural Network (DNN) computation. While scalable, such development tends to produce DL code that is error-prone, non-intuitive, and difficult to debug. Consequently, more natural, less error-prone imperative DL frameworks encouraging eager execution have emerged at the expense of run-time performance. While hybrid approaches aim for the "best of both worlds," the challenges in applying them in the real world are largely unknown. We conduct a data-driven analysis of challenges---and resultant bugs---involved in writing reliable yet performant imperative DL code by studying 250 open-source projects, consisting of 19.7 MLOC, along with 470 and 446 manually examined code patches and bug reports, respectively. The results indicate that hybridization: (i) is prone to API misuse, (ii) can result in performance degradation---the opposite of its intention, and (iii) has limited application due to execution mode incompatibility. We put forth several recommendations, best practices, and anti-patterns for effectively hybridizing imperative DL code, potentially benefiting DL practitioners, API designers, tool developers, and educators.
How analogue intelligence complements AI
How analogue intelligence complements AIPaul Rowe
Artificial Intelligence is providing benefits in many areas of work within the heritage sector, from image analysis, to ideas generation, and new research tools. However, it is more critical than ever for people, with analogue intelligence, to ensure the integrity and ethical use of AI. Including real people can improve the use of AI by identifying potential biases, cross-checking results, refining workflows, and providing contextual relevance to AI-driven results.
News about the impact of AI often paints a rosy picture. In practice, there are many potential pitfalls. This presentation discusses these issues and looks at the role of analogue intelligence and analogue interfaces in providing the best results to our audiences. How do we deal with factually incorrect results? How do we get content generated that better reflects the diversity of our communities? What roles are there for physical, in-person experiences in the digital world?
Challenges in Migrating Imperative Deep Learning Programs to Graph Execution:...
Challenges in Migrating Imperative Deep Learning Programs to Graph Execution:...Raffi Khatchadourian
Basic Descriptive Statistics
- 1. Basic Descriptive Statistics Mr. Siko Clarkston HS
- 2. Why? Descriptive statistics do just that: Describe Data! What we’ll cover in this slidecast – Mean (average) – Median – Mode – Range
- 3. Mean Fancy Formula What this means: add µ = ΣX/N up all your data, then divide by the number of data points
- 4. Mean Sample data: How to calculate: 98cm 76cm 98+76+82+54+90 = 82cm 400cm 54cm 90cm 400cm/5 = 80cm
- 5. Median The median is the middle data point in a set To determine the median, sort the data from smallest to largest and find the middle data point
- 6. Median Sample data: Rearranged Data: 98cm 54cm 76cm 76cm 82cm 82cm 54cm 90cm 90cm 98cm
- 7. Median If there is an even number of data, there will be two middle points. To find the median, take the average of those two data.
- 8. Median Sample Data: Rearranged Data: 4ml 2ml 8ml 4ml 12ml 8ml 2ml 12ml 4 + 8 = 12ml 12/2 = 6ml
- 9. Mode The mode is the most frequently occurring data point. To find the mode, arrange the data from smallest to largest, and then determine which amount occurs most often.
- 10. Mode Sample Data: Rearranged Data: 20g 23g 20g 20g 20g 30g 30g 22g 22g 27g 23g 23g 23g 23g 25g 20g 24g 23g 24g 25g 25g 23g 25g 27g 20g 23g 30g 30g
- 11. Range The range is the distance between the smallest and largest data point. To calculate, determine the smallest data point and the largest data point, then subtract the smallest from the largest.
- 12. Range Sample data: Rearranged Data: 98cm 54cm 76cm 76cm 82cm 82cm 54cm 90cm 90cm 98cm 98cm – 54cm = 44cm
- 13. Recap Mean, Median, Mode, and Range “describe” the data.
- 14. Acknowledgements American Chemical Society. (2006). Chemistry in the community: ChemCom (5th ed). New York: W.H. Freeman