Measures of Central Tendency: Ungrouped Data

Measures of Central Tendency: Ungrouped Data

• Definition of Arithmetic Mean: The sum of the numerical values of each observation divided by the total number of observations.
• Formula: ΣX / N, where ΣX is the sum of the values of all observations and N is the total number of observations.
• Example: Calculate the mean of the monthly salaries of 10 employees.

Discrete Data

• Formula: Σfx / N, where f is the frequency for each variable x and N is the total frequency.
• Example: Calculate the mean of the data with frequency distribution.

Continuous Data

• Formula: Σfx / N, where x is the midpoint of various classes, f is the frequency for each class, and N is the total frequency.
• Example: Calculate the mean of the monthly sales of 200 firms.

Mode

• Definition: The value which occurs most frequently in the dataset.
• Formula: Not applicable for ungrouped data.
• Merits: Easy to understand and calculate, not affected by extreme values, and can be calculated in case of open-end interval.
• Demerits: Not based on all observations, affected by sampling fluctuation, and not capable of further algebraic treatment.

Measures of Variability: Ungrouped Data

• Range: The simplest measure of variability, calculated by subtracting the minimum value from the maximum value.
• Formula: Maximum value - Minimum value.
• Variance: The average squared deviation of each data point from the mean.
• Formula: 1/n * Σ(xi - x)², where xi is each individual data point, x is the mean, and n is the number of data points.
• Standard Deviation: The square root of the variance.
• Formula: √(Variance).

Measures of Central Tendency and Variability: Grouped Data

• Mean: Calculated using the midpoint of each class interval as the representative value.
• Formula: Σ(f × X) / N, where f is the frequency of each interval, X is the midpoint of each interval, and N is the total frequency.
• Median: Calculated using the formula: Median = L + (f2 / N - F) × w, where L is the lower boundary of the median class, f2 is the frequency of the median class, N is the total frequency, F is the cumulative frequency of the class before the median class, and w is the width of the median class interval.
• Mode: The class interval with the highest frequency.

Measures of Shape, Skewness

• Kurtosis: Measures the peakedness or flatness of a distribution.
• Formula: n × s⁴ / Σ(xi - x)⁴, where xi is each individual data point, x is the mean, s is the sample standard deviation, and n is the number of observations.
• Skewness: Measures the asymmetry of the distribution.
• Formula: (n-1) × s³ / Σ(xi - x)³, where xi is each individual data point, x is the mean, s is the sample standard deviation, and n is the number of observations.

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

• Skewness provides information about the tail of the distribution.
• The relationship between the mean, median, and mode can be used to determine the skewness of a distribution.### Skewness and Central Tendency
• In a positively skewed distribution, the mean is typically larger than the median because the positive skewness pulls the mean towards the higher values.
• In a negatively skewed distribution, the mean is typically smaller than the median because the negative skewness pulls the mean towards the lower values.
• The median is not affected by extreme values or outliers as much as the mean, making it a robust measure of central tendency, particularly in skewed distributions.

Skewness and Median

• In a positively skewed distribution, the median is usually greater than the mean since it is less influenced by extreme values in the right tail.
• In a negatively skewed distribution, the median is usually less than the mean since it is less influenced by extreme values in the left tail.

Skewness and Mode

• The mode is the value that occurs most frequently in the dataset.
• In a positively skewed distribution, the mode is typically less than the mean and median because the majority of values are concentrated on the left side of the distribution, and the tail extends to the right.
• In a negatively skewed distribution, the mode is typically greater than the mean and median because the majority of values are concentrated on the right side of the distribution, and the tail extends to the left.

Coefficient of Skewness and 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 is defined as the ratio of the third standardized moment to the cube of the standard deviation.
• For a sample, 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.
• A positive kurtosis indicates a relatively peaked distribution (heavy-tailed), whereas a negative kurtosis indicates a relatively flat distribution (light-tailed).
• For a sample, the kurtosis can be calculated as: Kurtosis = (n-1) × s4 / Σ(xi-x̄)4.