Measures of Central Tendency: Ungrouped Data

Measures of Central Tendency

• Mean: Average value of a dataset, calculated by summing all values and dividing by the number of observations.
  • Ungrouped data: Mean = (∑x) / n
  • Grouped data: Mean = (∑fx) / n
• Median: Middle value of a dataset when arranged in order, divides the distribution into two equal parts.
  • Ungrouped data: Find the middle value if the number of observations is odd, or the average of the two middle values if the number of observations is even.
  • Grouped data: Median = l1 + ((N/2 - cf) * h) / f
• Mode: Most frequently occurring value in a dataset.
  • Ungrouped data: Find the value with the highest frequency.
  • Grouped data: Mode = l1 + ((f1 - f0) / (2*f1 - f0 - f2)) * h

Measures of Variability

• Range: Simplest measure of variability, calculated by subtracting the minimum value from the maximum value.
  • Range = Maximum value - Minimum value
• Variance: Measures the average squared deviation of each data point from the mean.
  • Variance = 1/n * ∑(xi - x̄)²
• Standard Deviation: Square root of the variance, measures the typical distance between each data point and the mean.
  • Standard Deviation = √Variance

Calculating Interquartile Range (IQR)

• IQR: Measures the spread of the middle 50% of the dataset.
• Steps:
  1. Order the data in ascending order.
  2. Find the first quartile (Q1): median of the lower half of the dataset.
  3. Find the third quartile (Q3): median of the upper half of the dataset.
  4. Calculate the IQR: IQR = Q3 - Q1

Calculating Mean Absolute Deviation (MAD)

• MAD: Measures the average absolute difference between each data point and the mean.
• Steps:
  1. Calculate the mean of the dataset.
  2. Calculate the absolute deviation of each data point from the mean.
  3. Calculate the average of the absolute deviations.

Coefficient of Skewness and Kurtosis

• Coefficient of Skewness: Measures the asymmetry of a probability distribution.
  • Coefficient of Skewness = 3(Mean - Median) / Standard Deviation
• Kurtosis: Measures the peakedness or flatness of a probability distribution.
  • Kurtosis = (n * (n + 1)) / ((n - 1) * (n - 2) * (n - 3)) * ∑(xi - x̄)⁴ / (σ⁴)

Percentiles

• Percentile: A value below which a certain percentage of the data falls.

• Steps to calculate a percentile:

  1. Sort the data in ascending order.
  2. Calculate the rank: R = P/100 * (n + 1)
  3. Interpolate if the rank is not an integer.
  4. Identify the value corresponding to the rank.### Measures of Central Tendency and Variability

• Mean: represents the average value of a dataset, calculated as the sum of all data points divided by the total number of data points (n)

• Standard Deviation: measures the amount of variation or dispersion of a set of values, expressed in the same units as the data, calculated as the square root of the variance

• Standard Deviation indicates how much individual data points typically differ from the mean; a larger standard deviation means data points are spread out over a wider range, while a smaller standard deviation means they are closer to the mean

Measures of Central Tendency and Variability for Grouped Data

• Mean: calculated using the midpoint of each class interval as the representative value, multiplied by the frequency of that interval, summed up, and divided by the total frequency of all the intervals
• Median: calculated using the formula: Median=L+(f2N‒F?)×w, where L = lower boundary of the median class, N = total frequency, F = cumulative frequency of the class before the median class, f = frequency of the median class, and w = width of the median class interval
• Mode: the class interval with the highest frequency
• Range: the difference between the highest and lowest values in the dataset
• Interquartile Range (IQR): calculated using the cumulative frequency distribution, finding the quartiles (Q1 and Q3) and then calculating the difference between them
• Variance and Standard Deviation: approximated using the midpoint of each class interval as the representative value and computing the variance and standard deviation based on these midpoints and their frequencies

Measures of Shape and Skewness

• Kurtosis: measures the peakedness or flatness of a distribution compared to a normal distribution, calculated using the formula: Kurtosis=n×s4‒i=1n(xi‒x?)4×fi‒3
• Skewness: measures the asymmetry of the distribution, calculated using the formula: Skewness=(n‒1)×s3‒i=1n(xi‒x?)3‒
• Skewness of 0 indicates a symmetric distribution, positive skewness means the tail on the right side of the distribution is longer or fatter, and negative skewness means the left tail is longer or fatter
• Skewness is related to the relationship of the mean, median, and mode, with skewness affecting the relative positions of these measures