Descriptive Measures-Measures of Positionality

Discussion on some descriptive tools identifying locations of data in a given data set

1. MEASURES OF POSITION Descriptive Statistics

2. OBJECTIVES Recognize, describe, calculate, and interpret the measures of location of data.

3. Measures of Location also known as measures of relative standing or measures of spread, are statistical tools used to determine where a particular data point falls within a dataset. These measures help assess the position or ranking of a value within the data distribution

4. COMMON MEASURES OF LOCATION Quartiles, Deciles and Percentiles Kurtosis Skewness

5. COMMON MEASURES OF LOCATION Quartiles , Percentiles, Percentile Rank

6. common measures of location are  quartiles  and  percentiles QUARTILE Quartiles  are numbers that separate the data into quarters (four parts).  Like the median, quartiles may or may not be an actual value in the set of data Quartiles divide the data into three points: Q1 (25th percentile), Q2 (50th percentile, or median), and Q3 (75th percentile).

7. Calculation Q1 : The value at which 25% of the data falls below and 75 % above . Q2 : The median, where 50% of the data is below and 50% above. Q3 : The value at which 75% of the data falls below and 25 % above. QUARTILE

8. Median (Q2) : The median, also known as the second quartile, is the middle value in the dataset. It is a measure of central tendency that is less sensitive to extreme outliers than the mean. Q2 divides the data into two equal halves. Explanation and Interpretation: First Quartile (Q1) : Q1, the 25th percentile, marks the point below which 25% of the data falls. It represents the lower end of the lower half of the dataset, providing information about the spread of the lower values.

9. Third Quartile (Q3) : Q3, the 75th percentile, marks the point below which 75% of the data falls. It represents the upper end of the upper half of the dataset, providing information about the spread of the higher values. Explanation and Interpretation:

10. Business Applications Sales Performance : Quartiles can be used to assess the performance of sales representatives by dividing them into quartiles based on their sales figures. This helps identify top performers (Q4), average performers (Q2 and Q3), and those who may need improvement (Q1). Employee Performance : Quartiles can be used to assess and rank employee performance based on various metrics such as sales figures, customer satisfaction scores, or project completion times. Employees in the top quartile may be recognized or rewarded, while those in the bottom quartile may receive additional training or coaching.

11. Business Applications Product Performance : Quartiles can help analyze the performance of different products or product categories. For instance, by categorizing products into quartiles based on sales revenue, businesses can identify their top-selling products and focus marketing efforts or inventory management accordingly. Customer Satisfaction : Quartiles can be used to analyze customer survey data to identify areas of improvement. For example, if customer satisfaction scores are divided into quartiles, businesses can focus on addressing issues that are most prevalent in the lowest quartile.

12. Considerations in Computing Quartiles Quartile.Exc calculates quartiles using exclusive boundaries. This means that when calculating Q1.Exc, it doesn't include the median value in the lower 25% of the data. Similarly, for Q3.Exc, it doesn't include the median value in the upper 25% of the data. Quartile.Inc calculates quartiles using inclusive boundaries. This means that when calculating Q1.Inc, it includes the median value in the lower 25% of the data. Similarly, for Q3.Inc, it includes the median value in the upper 25% of the data.

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14. PERCENTILE

15. Percentiles are statistical measures that divide a dataset into 100 equal parts, with each part representing a percentage of the data. PERCENTILE They help you understand how a specific data point compares to the entire dataset in terms of its relative position Percentiles are often expressed as a percentage and are commonly used to summarize and interpret data.

16. Sort the Data : Arrange your dataset in ascending or descending order. Calculation of Percentiles : Determine the Rank : Identify the rank (position) of the desired percentile. For example, if you want to find the 25th percentile (Q1), this represents the value below which 25% of the data falls.

17. Calculation of Percentiles : Interpolate If Necessary : If the rank is not an integer (e.g., you're looking for the 75th percentile, which falls between the 75th and 76th data points), you may need to interpolate to estimate the exact value. Calculate the Percentile : Once you have the rank or interpolated rank, you can find the corresponding value in the sorted dataset. This value is the desired percentile.

18. Median (50th Percentile) : The median is the middle value in a dataset. It divides the data into two equal halves, with 50% of the data below and 50% above it. The median is a measure of central tendency and is less affected by extreme outliers compared to the mean. Common Percentiles and Interpretation: Quartiles (25th, 50th, and 75th Percentiles) : Quartiles divide the data into four equal parts. The first quartile (Q1) is the 25th percentile, the second quartile (Q2, which is also the median) is the 50th percentile, and the third quartile (Q3) is the 75th percentile. They are used to analyze the spread of data and identify outliers.

19. Percentile Ranks (e.g., 90th Percentile) : Percentile ranks help you understand how a specific data point compares to the rest of the dataset. For example, if a student scores in the 90th percentile on a standardized test, it means they scored higher than 90% of the test-takers. Common Percentiles and Interpretation:

20. Application in Business Statistics : Performance Benchmarking: Businesses can use percentiles to set performance benchmarks. For instance, a company might aim to have customer service response times in the top 10th percentile to ensure exceptional service quality. Risk Assessment : In finance, percentiles are used to assess the risk associated with investments. For example, the 5th percentile of returns on an investment portfolio represents the lowest 5% of possible returns, helping investors understand potential losses.

21. Application in Business Statistics : Market Research : Percentiles are used to segment markets based on income, spending habits, or other factors. Companies can target specific percentile groups with tailored marketing strategies. Customer Segmentation : Percentiles can help businesses identify and categorize their most valuable customers based on factors like purchase frequency, transaction value, or loyalty.

22. Application in Business Statistics : Supply Chain Optimization : Companies can use percentiles to analyse the variability in supply chain lead times and manage inventory levels more effectively. Quality Control : In manufacturing, percentiles are used to ensure product quality. Products falling below a certain percentile may be considered defective and subject to further inspection or rejection.

23. Considerations in Computing Percentile Percentile.Exc calculates percentiles using exclusive boundaries. It does not include the specified percentile value itself in the calculated percentage of data below that percentile. Percentile.Inc : Percentile.Inc calculates percentiles using inclusive boundaries. It includes the specified percentile value itself in the calculated percentage of data below that percentile.

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25. SKEWNESS

26. Skewness an important concept in statistics that helps us understand the shape and symmetry of a data distribution measure of the asymmetry of a probability distribution. tells us whether the data is concentrated more on one side of the mean than the other Skewness is a crucial concept in descriptive statistics, especially when analyzing datasets in fields like business, finance, economics, and social sciences.

27. TYPES OF SKEWNESS Positive Skewness (Right Skewed): In a positively skewed distribution, the tail on the right-hand side (greater values) is longer or fatter than the left tail. This means that the data has outliers on the right side and is concentrated towards the left.

28. TYPES OF SKEWNESS Negative Skewness (Left Skewed): In a negatively skewed distribution, the tail on the left-hand side (smaller values) is longer or fatter than the right tail. This indicates that the data has outliers on the left side and is concentrated towards the right.

29. COMPUTATION OF SKEWNESS The most common measure of skewness is the skewness coefficient, denoted as "γ" or "Skew." There are different formulas to compute skewness, but one of the commonly used formulas is Pearson's First Coefficient of Skewness (Skewness coefficient ):

30. COMPUTATION OF SKEWNESS The skewness coefficient can be positive, negative, or zero, indicating the direction and degree of skewness. Positive skewness indicates right skew (tail to the right). Negative skewness indicates left skew (tail to the left). Zero skewness indicates a perfectly symmetrical distribution.

31. Interpreting Skewness: The skewness coefficient can be positive, negative, or zero, indicating the direction and degree of skewness. Greater than +1 or less than -1 suggests a highly skewed distribution. Between -1 and +1 suggests a moderately skewed distribution. Close to 0 suggests a nearly symmetrical distribution.

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