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Comprehensive Guide to Descriptive Statistics and Data Analysis, Lecture notes of Data Analysis & Statistical Methods

University of DerbyData Analysis & Statistical Methods

This lectures note provides a clear and concise understanding of descriptive statistics, making it essential for students studying business analytics or quantitative methods. It explores critical concepts and easy-to-follow explanations, including: Measures of Central Tendency: Mean, Median, Mode, and their significance. Measures of Variation: Range, Variance, and Standard Deviation for analyzing data spread. Distribution Shape Analysis: Skewness, Z-scores, and identifying outliers. Quartiles and Interquartile Range (IQR): For understanding data spread. Population Parameters vs. Sample Statistics: Learn the differences and practical applications. Perfect for business students aiming to excel in quantitative analysis and data interpretation Textbook used to integrate these lecture notes with: Levine, D, Stephan, D, & Szabat, K 2020, Statistics for Managers Using Microsoft Excel, Global Edition, Pearson Education, Limited, Harlow. Available from: ProQuest Ebook Central

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2023/2024

Available from 12/19/2024

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Quantitative Skills for Business- Lecture 3: Descriptive Measures and Variables
Central Tendency:
• Extent to which values of a numerical variable group around a central value.
Variation:
• The amount of scattering from a central value that the values of a numerical
variable show.
Shape:
• The pattern of distribution of values from the lowest to the highest.
Measures of Central Tendency
The Mean:
• Definition: The sum of all values divided by the number of values.
• Effect of Outliers: The mean can be significantly affected by extreme values
(outliers).
• Calculation Example:
o For the set {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, the mean =
(11+12+13+14+15+16+17+18+19+20) / 10 = 155 / 10 = 15.5.
The Median:
• Definition: The middle value when data is ordered from smallest to largest.
• Odd Data Set: The median is the middle value.
• Even Data Set: The median is the average of the two middle values.
• Formula: Position of the median = (n+1)/2, where n is the number of
observations.
• Calculation Example:
o For {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, the median is (15+16)/2 = 15.5.
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Download Comprehensive Guide to Descriptive Statistics and Data Analysis and more Lecture notes Data Analysis & Statistical Methods in PDF only on Docsity!

Quantitative Skills for Business- Lecture 3: Descriptive Measures and Variables Central Tendency:

  • Extent to which values of a numerical variable group around a central value. Variation:
  • The amount of scattering from a central value that the values of a numerical variable show. Shape:
  • The pattern of distribution of values from the lowest to the highest. Measures of Central Tendency The Mean:
  • Definition: The sum of all values divided by the number of values.
  • Effect of Outliers: The mean can be significantly affected by extreme values (outliers).
  • Calculation Example: o For the set {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, the mean = (11+12+13+14+15+16+17+18+19+20) / 10 = 155 / 10 = 15.5. The Median:
  • Definition: The middle value when data is ordered from smallest to largest.
  • Odd Data Set: The median is the middle value.
  • Even Data Set: The median is the average of the two middle values.
  • Formula: Position of the median = (n+1)/2, where n is the number of observations.
  • Calculation Example: o For {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, the median is (15+16)/2 = 15.5.

The Mode:

  • Definition: The value that occurs most frequently.
  • Characteristics: o There may be no mode or more than one mode. o The mode is not affected by outliers. Geometric Mean:
  • Definition: The nth root of the product of n values. Used mainly for data sets involving growth rates (e.g., financial returns).

o Coefficient of Variation (CV):

  • Definition: A percentage measure of the relative variability of the data set compared to the mean.
  • Formula:
  • Usage: Useful for comparing variability between different data sets, especially if they are in different units. Z-Score (for Outliers):
  • Definition: The number of standard deviations a data value is from the mean.
  • Formula:
  • Outliers: o If Z<−3Z < - 3Z<−3 or Z>3Z > 3Z>3, the data point is considered an extreme outlier. o The larger the absolute value of the Z-score, the farther the value is from the mean. Shape of Distribution Skewness:
  • Definition: Measures the asymmetry of the data. o Symmetric Distribution: Mean = Median. o Left-Skewed (Negative Skew): Mean < Median. o Right-Skewed (Positive Skew): Mean > Median.
  • Values: o Skewness ranges between - 4 and +4. o Close to zero = symmetrical. o Far from zero = highly skewed. Quartile Measures Quartiles:
  • Q1: The first quartile (25th percentile).
  • Q2: The second quartile, or the median (50th percentile).
  • Q3: The third quartile (75th percentile). Interquartile Range (IQR):
  • Definition: Measures the spread of the middle 50% of the data.
  • Formula: IQR = Q3 - Q1.

Population Parameters:

  • Definition : Population parameters are descriptive measures used to summarize and describe characteristics of an entire population.
  • Notation : Greek letters (e.g., μ, σ², σ) are typically used to represent population parameters.
  • Key Population Parameters :
    1. Population Mean (μ) : ▪ Interpretation : The mean gives the average value of the population.
    2. Population Variance (σ²) :
    3. ▪ Interpretation : The variance measures how much individual values in the population differ from the mean (i.e., the spread of the data).
  1. Population Standard Deviation (σ) : ▪ ▪ Interpretation : The standard deviation is the square root of the variance and is in the same units as the original data. It gives a direct sense of the average distance from the mean. Sample Statistics:
  • Definition : Sample statistics are measures used to describe and summarize characteristics of a subset of the population (the sample). These are used to estimate population parameters.
  • Notation : Latin letters are typically used to represent sample statistics.
  • Key Sample Statistics :
  1. Sample Mean ▪ Interpretation : It represents the average value of the sample.
  2. Sample Variance : ▪ Interpretation : Sample variance is the average of the squared deviations from the sample mean, used to estimate the population variance.
  3. Sample Standard Deviation : ▪ Interpretation : The square root of the sample variance, which is in the same units as the original data. It gives a sense of how spread out the sample data is.
  1. Population Mean vs. Sample Mean : o Population Mean : If you survey every student in a university and calculate the average age, that’s the population mean (μ). o Sample Mean : If you only survey 100 students from the university and calculate the average age, that’s the sample mean.
  2. Population Variance vs. Sample Variance : o Population Variance : If you know the exact ages of all students in the university and compute how much they vary from the population mean, you’re calculating the population variance. o Sample Variance : If you only have the ages of 100 students and compute the variance from their mean, you’re calculating the sample variance. Summary:
  • Population Parameters are exact values describing the entire population, but they are often difficult or impossible to calculate because it requires complete data for the entire population.
  • Sample Statistics are used to estimate population parameters and are subject to variability depending on the sample.
  • Bessel’s Correction (n-1) ensures that the sample variance is an unbiased estimator of the population variance, which is especially important when making inferences from samples.