Stat 3

• Measures of Central Tendency:

  • Mean: arithmetic average calculated by summing up all values and dividing by the number of observations
  • Median: middle value of a dataset when arranged in ascending or descending order
  • Mode: most frequently occurring value in a dataset
• Measures of Central Tendency provide a single, representative value that summarizes the center or typical value of a dataset
• Common measures of central tendency are mean, median, and mode
• Mean is affected by extreme values (outliers) and is a good measure when the data set is normally distributed
• Median is less sensitive to extreme values and is a good measure when the data set has outliers
• Mode is not affected by extreme values and can be used for numerical or categorical data
• Measures of Variation:

  • Variance: average of squared deviations of values from the mean
  • Standard Deviation: square root of the variance
• Measures of Variation provide information on the spread, consistency, variability, or dispersion of the data values
• Common measures of variation include range, variance, and standard deviation
• Range is the simplest measure of variation, calculated as the difference between the largest and smallest values
• Variance and Standard Deviation show variation about the mean and have the same units as the original data
• Measures of Relative Position:

  • Quartiles: split the ranked data into 4 segments with an equal number of values per segment
  • Percentiles: divide the data into 100 equal parts, each representing a percentage
• Measures of Relative Position describe the location of a particular data point within a dataset relative to other data points
• Quartiles include Q1, Q2 (median), and Q3, with the Interquartile Range (IQR) measuring the spread in the middle 50% of the data
• Percentiles divide the data into 100 equal parts, indicating the point below which a certain percentage of data falls
• Box-and-whiskers plot visually displays the distribution of a dataset, highlighting the five-number summary and any potential outliers
• Boxplot includes the smallest value, Q1, median (Q2), Q3, and largest value, showing the distribution shape of the data
• Shape of a Distribution

  • Describes how data are distributed

  • Two useful shape related statistics are:

  ➢Skewness

  ✓Measures the extent to which data values are not symmetrical

  ➢Kurtosis

  ✓Kurtosis affects the peakedness of the curve of the

  distribution—that is, how sharply the curve rises approaching

  the center of the distribution
• Outlier: An observation that lies far from most of the others in a sample or population
• Measures of Variation

  • Provide information on the spread or consistency or variability or

  dispersion of the data values.
• Why The Range Can Be Misleading

  • Does not account for how the data are distributed

  • Sensitive to outliers
• Coefficient of Variation

  • Measures relative variation

  • Always in percentage (%)

  • Shows variation relative to mean

  • Can be used to compare the variability of two or more sets of data

  measured in different units
• Quartile Measures:

  The Interquartile Range (IQR)

  • The IQR measures the spread in the middle 50% of the data

  • The IQR is also called the midspread because it covers the middle 50%

  of the data

  • The IQR is a measure of variability that is not influenced by outliers or

  extreme values
• Shape of Boxplots

  • If data are symmetric around the median then the box and central line

  are centered between the endpoints

  • A Boxplot can be shown in either a vertical or horizontal orientation