Central Tendency, Position, and Dispersion

• Measures of central tendency, measures of position, and measures of dispersion are statistical tools used to analyze and summarize data sets, both ungrouped and grouped.

Measures of Central Tendency

• Central tendency aims to determine a single value that accurately represents the center of the data.
• Common measures include the mean, median, and mode.

Mean (Ungrouped Data)

• The mean, also known as the average, is calculated by summing all the values in the data set and dividing by the number of values.
• Formula: Mean = (Sum of all values) / (Number of values)
• Example: For the data set [3, 6, 7, 8, 11], the mean is (3 + 6 + 7 + 8 + 11) / 5 = 7.

Median (Ungrouped Data)

• The median is the middle value in a data set that is arranged in ascending or descending order.
• If there is an even number of values, the median is the average of the two middle values.
• Example: For the data set [3, 6, 7, 8, 11], the median is 7. For [3, 6, 7, 8, 11, 15], the median is (7 + 8) / 2 = 7.5.

Mode (Ungrouped Data)

• The mode is the value that appears most frequently in a data set. A data set can have no mode, one mode (unimodal), or multiple modes (bimodal, trimodal, etc.).
• Example: For the data set [2, 3, 6, 6, 7, 8, 11], the mode is 6.

Mean (Grouped Data)

• For grouped data (data presented in frequency tables), the mean is calculated using the formula: Mean = Σ(fᵢ * xᵢ) / Σfᵢ, where fᵢ is the frequency of each class and xᵢ is the midpoint of each class.
• Example: Consider a frequency table with classes 1-5 (f=4), 6-10 (f=8), and 11-15 (f=3). The midpoints are 3, 8, and 13, respectively. The mean is (43 + 88 + 3*13) / (4 + 8 + 3) = 97 / 15 ≈ 6.47.

Median (Grouped Data)

• The median for grouped data is found using the formula: Median = L + [(N/2 - CF) / f] * w, where L is the lower boundary of the median class, N is the total frequency, CF is the cumulative frequency of the class before the median class, f is the frequency of the median class, and w is the class width.
• Example: Using the frequency table from above, if N = 15, N/2 = 7.5. The median class is 6-10 (since the cumulative frequency up to 5 is 4, and up to 10 is 12). L = 5.5, CF = 4, f = 8, and w = 5. Median = 5.5 + [(7.5 - 4) / 8] * 5 ≈ 7.69.

Mode (Grouped Data)

• The mode for grouped data is estimated using the formula: Mode = L + [(fₘ - f₁) / (2fₘ - f₁ - f₂)] * w, where L is the lower boundary of the modal class, fₘ is the frequency of the modal class, f₁ is the frequency of the class before the modal class, f₂ is the frequency of the class after the modal class, and w is the class width.
• Example: Using the frequency table from above, the modal class is 6-10. L = 5.5, fₘ = 8, f₁ = 4, f₂ = 3, and w = 5. Mode = 5.5 + [(8 - 4) / (2*8 - 4 - 3)] * 5 ≈ 7.9.

Measures of Position

• Measures of position describe the location of a specific data value within the distribution.
• Common measures include quartiles, deciles, and percentiles.

Quartiles (Ungrouped Data)

• Quartiles divide a data set into four equal parts. The first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the 50th percentile (median), and the third quartile (Q3) is the 75th percentile.
• To find quartiles, arrange the data in ascending order and find the median (Q2). Then find the median of the lower half (Q1) and the median of the upper half (Q3).
• Example: For the data set [3, 6, 7, 8, 11, 15, 20, 21], Q2 (median) = (8 + 11) / 2 = 9.5. The lower half is [3, 6, 7, 8], so Q1 = (6 + 7) / 2 = 6.5. The upper half is [11, 15, 20, 21], so Q3 = (15 + 20) / 2 = 17.5.

Deciles (Ungrouped Data)

• Deciles divide a data set into ten equal parts. Each decile represents 10% of the data.
• To find deciles, arrange the data in ascending order and determine the value corresponding to the desired percentage. For example, the 3rd decile (D3) is the value below which 30% of the data falls.
• Example: In a data set of 100 values, D3 would be the 30th value when the data is sorted.

Percentiles (Ungrouped Data)

• Percentiles divide a data set into one hundred equal parts. Each percentile represents 1% of the data.
• To find percentiles, arrange the data in ascending order and determine the value corresponding to the desired percentage. For example, the 60th percentile (P60) is the value below which 60% of the data falls. -Example: In a data set of 100 values, P60 would be the 60th value when the data is sorted.

Quartiles (Grouped Data)

• For grouped data, quartiles are calculated using the formula: Qₖ = L + [(k(N/4) - CF) / f] * w, where k is the quartile number (1, 2, or 3), L is the lower boundary of the quartile class, N is the total frequency, CF is the cumulative frequency of the class before the quartile class, f is the frequency of the quartile class, and w is the class width.
• Example: Using the frequency table from before (classes 1-5 (f=4), 6-10 (f=8), and 11-15 (f=3)), to find Q1, k = 1, N = 15, so N/4 = 3.75. The Q1 class is 1-5. L = 0.5, CF = 0, f = 4, and w = 5. Q1 = 0.5 + [(3.75 - 0) / 4] * 5 ≈ 5.19.

Deciles (Grouped Data)

• For grouped data, deciles are calculated using the formula: Dₖ = L + [(k(N/10) - CF) / f] * w, where k is the decile number (1 to 9), L is the lower boundary of the decile class, N is the total frequency, CF is the cumulative frequency of the class before the decile class, f is the frequency of the decile class, and w is the class width.
• Example: Using the same frequency table, to find D3, k = 3, N = 15, so 3N/10 = 4.5. The D3 class is 6-10 (cumulative frequency up to 5 is 4). L = 5.5, CF = 4, f = 8, and w = 5. D3 = 5.5 + [(4.5 - 4) / 8] * 5 ≈ 5.81.

Percentiles (Grouped Data)

• For grouped data, percentiles are calculated using the formula: Pₖ = L + [(k(N/100) - CF) / f] * w, where k is the percentile number (1 to 99), L is the lower boundary of the percentile class, N is the total frequency, CF is the cumulative frequency of the class before the percentile class, f is the frequency of the percentile class, and w is the class width.
• Example: Using the same frequency table, to find P60, k = 60, N = 15, so 60N/100 = 9. The P60 class is 6-10 (cumulative frequency up to 5 is 4, up to 10 is 12). L = 5.5, CF = 4, f = 8, and w = 5. P60 = 5.5 + [(9 - 4) / 8] * 5 ≈ 8.63.

Measures of Dispersion

• Measures of dispersion describe the spread or variability of data points in a data set.
• Common measures include range, variance, and standard deviation.

Range (Ungrouped Data)

• The range is the difference between the maximum and minimum values in a data set.
• Formula: Range = Maximum value - Minimum value
• Example: For the data set [3, 6, 7, 8, 11], the range is 11 - 3 = 8.

Variance (Ungrouped Data)

• The variance measures the average squared deviation of each data point from the mean.
• Formula: Variance = Σ(xᵢ - μ)² / N, where xᵢ is each data point, μ is the mean, and N is the number of data points. For a sample variance, the formula is Σ(xᵢ - x̄)² / (n-1), where x̄ is the sample mean and n is the sample size.
• Example: For the data set [3, 6, 7, 8, 11], the mean is 7. The variance is [(3-7)² + (6-7)² + (7-7)² + (8-7)² + (11-7)²] / 5 = (16 + 1 + 0 + 1 + 16) / 5 = 34 / 5 = 6.8. Sample variance would be 34/4 = 8.5.

Standard Deviation (Ungrouped Data)

• The standard deviation is the square root of the variance. It measures the spread of data around the mean.
• Formula: Standard Deviation = √Variance
• Example: For the data set [3, 6, 7, 8, 11] with a variance of 6.8, the standard deviation is √6.8 ≈ 2.61. Sample standard deviation ≈ √8.5 ≈ 2.92.

Range (Grouped Data)

• For grouped data, the range is estimated as the difference between the upper boundary of the highest class and the lower boundary of the lowest class.
• Example: Using the frequency table from before (classes 1-5, 6-10, and 11-15), the range is 15.5 - 0.5 = 15.

Variance (Grouped Data)

• For grouped data, the variance is calculated using the formula: Variance = Σ[fᵢ * (xᵢ - μ)²] / Σfᵢ, where fᵢ is the frequency of each class, xᵢ is the midpoint of each class, and μ is the mean of the grouped data. For a sample variance, the denominator is Σfᵢ - 1.
• Example: Using the frequency table and calculated mean (6.47), the variance is calculated by summing the squared differences between each midpoint and the mean, weighted by frequency, and then divided by the total frequency.

Standard Deviation (Grouped Data)

• The standard deviation for grouped data is the square root of the variance.
• Formula: Standard Deviation = √Variance
• Example: For the grouped data with a calculated variance, take the square root of the variance to find the standard deviation.