Measures of Central Tendency: Ungrouped Data

Measures of Central Tendency and Variability

• Standard Deviation:
  • Measures how much individual data points differ from the mean
  • Calculated as: n * (sum of (xi - x)^2)
  • Larger standard deviation means data points are spread out over a wider range, while smaller standard deviation means they are closer to the mean
• Measures of Central Tendency and Variability for Grouped Data:
  • Mean: calculated using midpoint of each class interval and frequency
  • Median: calculated using cumulative frequency distribution
  • Mode: class interval with the highest frequency
  • Range: difference between highest and lowest values
  • Interquartile Range (IQR): difference between third quartile (Q3) and first quartile (Q1)
  • Variance and Standard Deviation: approximated using midpoint of each class interval and frequency

Skewness and Measures of Shape

• Kurtosis:
  • Measures the peakedness or flatness of a distribution
  • Calculated as: n * (sum of (xi - x)^4) / (s^4)
  • Positive kurtosis indicates a relatively peaked distribution, while negative kurtosis indicates a relatively flat distribution
• Skewness:
  • Measures the asymmetry of the distribution
  • Calculated as: (n-1) * (sum of (xi - x)^3) / (s^3)
  • Positive skewness means the tail on the right side of the distribution is longer or fatter, while negative skewness means the left tail is longer or fatter

Relationship between Skewness and Mean, Median, and Mode

• Skewness and Mean:
  • Skewness provides information about the tail of the distribution
  • Measures of Central Tendency (Mean, Median, Mode) are related to skewness
• Measures of Central Tendency: Ungrouped Data:
  • Mean: calculated as sum of values divided by total number of observations
  • Median: middle value of the dataset when arranged in order
  • Mode: most frequently occurring value

Measures of Central Tendency: Discrete and Continuous Data

• Discrete Data:
  • Mean: calculated as sum of values multiplied by frequency, divided by total frequency
  • Median: calculated using cumulative frequency distribution
  • Mode: value with the highest frequency
• Continuous Data:
  • Mean: calculated using midpoint of each class interval and frequency
  • Median: calculated using cumulative frequency distribution
  • Mode: class interval with the highest frequency

Merits and Demerits of Mean, Median, and Mode

• Mean:
  • Merits: easy to understand and calculate, based on all observations, capable of further algebraic treatment
  • Demerits: highly affected by extreme values
• Median:
  • Merits: easy to understand and calculate, not affected by extreme values, located graphically
  • Demerits: not based on all observations, affected by sampling fluctuation
• Mode:
  • Merits: easy to understand and calculate, not affected by extreme values, located graphically
  • Demerits: not based on all observations, highly affected by sampling fluctuation, not capable of further algebraic treatment

Measures of Variability: Ungrouped Data

• Range:
  • Simplest measure of variability
  • Calculated as: maximum value - minimum value
• Variance:
  • Measures the average squared deviation of each data point from the mean
  • Calculated as: 1/n * sum of (xi - x)^2
• Standard Deviation:
  • Square root of the variance
  • Measures the typical distance between each data point and the mean
  • Calculated as: sqrt(variance)

Interquartile Range (IQR)

• Calculating IQR:
  • Order the data in ascending order
  • Find the first quartile (Q1) and third quartile (Q3)
  • Calculate the IQR as: Q3 - Q1
• IQR:
  • Measures the middle 50% of the data
  • Resistant to outliers
  • Calculated as: Q3 - Q1

Mean Absolute Deviation (MAD)

• Calculating MAD:
  • Calculate the mean of the dataset
  • Calculate the absolute difference between each data point and the mean
  • Calculate the mean of the absolute differences
• MAD:
  • Measures the average absolute difference between each data point and the mean
  • Calculated as: 1/n * sum of |xi - x|
  • Used to describe the spread of the data### Measures of Central Tendency
• Mean: the average value of a dataset, calculated by summing all values and dividing by the total number of observations
  • Formula: Mean = Σx / n
  • Example: monthly salary of 10 employees: Mean = (2500 + 2700 + ... + 2400) / 10 = 2530

Skewness and Mean

• In a positively skewed distribution, the mean is typically larger than the median
• In a negatively skewed distribution, the mean is typically smaller than the median

Skewness and Median

• In a positively skewed distribution, the median is usually greater than the mean
• In a negatively skewed distribution, the median is usually less than the mean

Coefficient of Skewness and Kurtosis

• Coefficient of Skewness: measures the asymmetry of a probability distribution
  • Formula: Coefficient of Skewness = 3(Mean - Median) / Standard Deviation
• Kurtosis: measures the peakedness or flatness of a probability distribution
  • Formula: Kurtosis = (n-1) * (sum((xi - x)^4) / (n * s^4))

Measures of Central Tendency (Cont.)

• Median: the middle value of a dataset when arranged in order
  • For odd number of observations: median is the middle value
  • For even number of observations: median is the average of the two middle values
• Mode: the most frequently occurring value in a dataset
  • Example: X = {3, 4, 5, 5, 6, 7, 8, 8, 8, 9} -> Mode = 8

Inter-Quartile Range (IQR)

• IQR: measures the spread of the middle 50% of a dataset
  • Formula: IQR = Q3 - Q1
  • Example: exam scores {65, 70, 75, 80, 85, 90, 95, 100} -> IQR = 87.5 - 72.5 = 15

Mean Absolute Deviation (MAD)

• MAD: measures the average distance of each data point from the mean
  • Formula: MAD = (1/n) * SUM(|xi - x|)
  • Example: exam scores {65, 70, 75, 80, 85, 90, 95, 100} -> MAD = 10

Variance and Standard Deviation

• Variance: measures the spread of a dataset
  • Formula: Variance = SUM((xi - x)^2) / n
• Standard Deviation: the square root of the variance
  • Formula: Standard Deviation = sqrt(Variance)

Calculating Percentiles

• Steps to determine the location of a percentile:

  1. Sort the data in ascending order
  2. Calculate the rank of the percentile
  3. Interpolate if the rank is not an integer
  4. Identify the value corresponding to the rank### Measures of Central Tendency and Variability: Grouped Data

• The mean of grouped data is calculated using the midpoint of each class interval, which is multiplied by the frequency of that interval, summed up, and divided by the total frequency.

• The median of grouped data is calculated using the formula: Median=L+(f2N–F)×w, where L is the lower boundary of the median class, N is the total frequency, F is the cumulative frequency of the class before the median class, f is the frequency of the median class, and w is the width of the median class interval.

• The mode of grouped data is the class interval with the highest frequency.

Measures of Shape and Skewness

• Kurtosis measures the peakedness or flatness of a distribution compared to a normal distribution, with positive kurtosis indicating a peaked distribution and negative kurtosis indicating a flat distribution.
• The formula for sample kurtosis is: Kurtosis=n×s4–3, where xi is the individual data point, x is the sample mean, fi is the frequency of each data point, n is the total number of observations, and s is the sample standard deviation.

Skewness and the Relationship of the Mean, Median, and Mode

• Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
• Positive skewness indicates a longer tail on the right side of the distribution, and negative skewness indicates a longer tail on the left side.
• The relationship between skewness and the mean, median, and mode is as follows:
  • Skewness and Mean: Skewness affects the mean, with positive skewness pulling the mean towards higher values and negative skewness pulling the mean towards lower values.
  • Skewness and Median: The median is less affected by extreme values and outliers, making it a robust measure of central tendency, particularly in skewed distributions.
  • Skewness and Mode: The mode is typically less than the mean and median in positively skewed distributions and greater than the mean and median in negatively skewed distributions.

Coefficient of Skewness, Kurtosis

• The coefficient of skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
• The coefficient of skewness can be calculated as: Coefficient of Skewness = 3(Mean–Median) / Standard Deviation.
• Kurtosis measures the peakedness or flatness of a probability distribution compared to a normal distribution.
• The kurtosis can be calculated as: Kurtosis=(n–1)×s4–3, where xi are the data points, x is the mean, s is the standard deviation, and n is the sample size.