1-Descriptive Statistics - pdf file descriptive
- 1. Statistical Methods for Decision Making Page 1
- 2. Agenda • Sampling • Basic Descriptive Statistics • Probability basics, Bayes Theorem • Probability distributions • Sampling Distribution • Interval Estimation and Hypothesis Testing • Introduction to Linear Regression Page 2
- 4. Agenda • Population and Sample • Data Collection • Types of data • Measures of central tendence • Measures of dispersion • Covariance and Coefficient of correlation Page 4
- 5. Population and Sample • The collection of all data points is the “population” or the “universe” data for a process • A subset of points drawn from a population is called “sample” • Measurement of a characteristic of population is called “parameter” • Measurement of a characteristic of sample is called “statistic” Page 5
- 6. Data Collection – Data sources • Primary vs Secondary data sources • Internal vs External data sources Page 6
- 7. Data Collection • Observation • Questionnaires and Surveys • Interviews etc Page 7
- 8. Data Collection - Sampling • Non-probability sampling: Selection is not statistically random. E.g. based on judgement or convenience. • Probability sampling • Random sampling - Every item has equal chance of being selected • Sampling without replacement • Samling with replacement • Stratified random sampling: Select randomly from predefined subgroups (strata) • Cluster sampling – Sampling from naturally occurring clusters, e.g. cities. • Systematic sampling – Divide into n groups containing k items. Randomly select from first k items. Then select every kth item. Page 8
- 9. Types of Data Page 9 Example: Number of items sold Example: Weight of a product Example: Preferred brand name, Gender Types of Data Categorical Numeric
- 10. Measurement Scale Nominal data does not have order. For example: gender Ordinal data has a meaningful order. For example: appraisal rating Page 10 Interval example: Temperature in Celsius. Ratio example: Cost of an item Measurement Scale Categorical (Qualitative) Numeric (Quantitative)
- 11. Descriptive Statistics • Central Tendency • Mean: Arithmetic mean of numbers. Add the observations and divide by count of the observations. Mean is affected by extreme values • Median: When observations are sorted in ascending order, the middle observation is median. If we have n observations, the (n+1)/2 th observation is median. The median can be an observation or between two observations • Mode: Mode is the most frequently occurring data point in a data set Page 11
- 12. Descriptive Statistics • Range: It is the difference between the maximum and minimum values in a data set. Affected by extreme values • Inter Quartile Range (IQR) – IQR is the distance between the first and the third quartile. • First quartile (Q1) has 25% observation lower than it. (i.e. 25th percentile) • Third quartile (Q3) has 75% observation lower than it • Median is also called second quartile (Q2) • Variance is measured as the average of sum of squared difference between each data point (represented by xi) and the mean represented by Page 12 n Σ (xi - ҧ 𝑥)2 i=1 ------------ n - 1 N Σ (xi - )2 i=1 ------------ N Unbiased formula
- 13. Descriptive Statistics • Standard deviation is one of the most popular measure of spread. It is the square root of the variance. Page 13 n Σ (xi - ҧ 𝑥)2 i=1 ------------ n - 1 N Σ (xi - )2 i=1 ------------ N Unbiased formula
- 14. Descriptive Statistics • Listing of Minimum, 1st quartile, Median, 3rd Quartile and Maximum is also called “five number summary” • Boxplot: A boxplot is a standardized way of displaying the distribution of data based on a five-number summary (“minimum”, first quartile (Q1), median, third quartile (Q3), and “maximum”). • The box is drawn from Q1 to Q3 • Each whisker can extend maximum of (1.5 * IQR) beyond Q1 and Q3 • Any points beyond whisker, called outliers, are also plotted Page 14
- 15. Discussion • How to interpret the following Page 15 OR B A A B
- 16. Descriptive Statistics • Histogram: A histogram is a visual representation of the underlying frequency distribution of a data attribute. • Height of bars represents the frequency of occurrence • Width of the bars is called class intervals Page 16
- 17. Skewness • A measure of the asymmetry of distribution of data • Types of Skewness: • Positive Skew (Right Skew): Tail on the right side is longer. • Negative Skew (Left Skew): Tail on the left side is longer. • Symmetrical Distribution: Skewness ≈ 0. Page 17 Image credit: https://www.analyticsvidhya.com/blog/2020/07/what-is-skewness-statistics/
- 18. Kurtosis • A measure of “tailedness” or “peakedness” of a data • Note: Additional information: • Mesokurtic: Normal distribution. Excess Kurtosis ≈ 0. • Leptokurtic: Peaked distribution with fat tails. Excess Kurtosis > 0. • Platykurtic: Flat distribution with thin tails. Excess Kurtosis < 0. Page 18
- 19. Coefficient of Variation Comparing dispersion – food for thought • Following is the performance of two factories in terms number of parts produced per day Factory 1: Standard deviation 10 Factory 2: Standard deviation 12 What is the observation? Possible answer: it appears that the factory-2 has more variation in output (note: this may not a correct answer) Page 19
- 20. Coefficient of Variation • Coefficient of variation is a type of relative measure of dispersion. • It is expressed as the ratio of the standard deviation to the mean. • Coefficient of variation = Standatd deviation Mean = 𝜎 𝜇 OR 𝑠 ҧ 𝑥 • This value tells you the size of the standard deviation relative to the mean. It is often expressed as percentage • Instead of standard deviation, the coefficient of variation should be used for comparison of variability between data sets on different scales or very different means Page 20
- 21. Coefficient of Variation • Following is the performance of two factories in terms number of parts produced per day Factory 1: Standard deviation = 10 Factory 2: Standard deviation = 12 Factory 1: Mean = 100 Factory 2: Mean = 200 Factory 1: Coefficient of variation = 10/100 = 0.1 = 10% Factory 2: Coefficient of variation = 12/200 = 0.06 = 6% Now, what is the observation? Factory-2 standard deviation is 6% of it’s mean while Factory-1 standard deviation is 10% of it’s mean. So relatively, Factory-1 has more variation in output Page 21
- 22. Covariance • Covariance measures the joint variability between two numerical variables (X and Y). • Covariance is calculated as • Covariance measures the extent to which two variables vary linearly • It reveals whether two variables move in the same or opposite directions. • The larger the X and Y values, the larger the covariance. A value doesn’t tell us exactly how strong that relationship is Page 22
- 23. Covariance • The sign of covariance reveals whether two variables move in the same or opposite directions. • The larger the X and Y values, the larger the covariance. The values of Covariance is not range bound. Covariance value doesn’t indicate how strong that relationship is Image: https://www.allmath.com/covariance.php Page 23 Positive covariance Negative covariance Near Zero covariance Positive covariance Negative covariance
- 24. Coefficient of Correlation • Coefficient of correlation, denoted as ‘r’ as calculated as: • Its value can range between -1 to +1 • The sign of coefficient of correlation tell the direction of relation • The value tells the measures the strength of a linear relationship between two variables (X and Y) • Note that the coefficient of correlation does not indicate causality Page 24
- 25. Coefficient of Correlation • A value closer to +1 indicates a strong positive (direct) relationship while a value closer to -1 indicates a strong negative (inverse) relationship • A value close to zero indicates no linear relationship Page 25
- 26. Coefficient of Correlation • Note that the coefficient of correlation does not indicate causality Credit: https://xkcd.com/552/ Page 26