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Showing posts from August, 2024

Range

Definition:      The range is the difference between the highest and lowest values in a dataset. Formula: Range = Maximum Value − Minimum Value Example: Given the dataset: [ 7 , 3 , 1 , 5 , 9 ] [7, 3, 1, 5, 9] Find the maximum and minimum values :   Max = 9 , Min = 1   \text{Max} = 9,\quad \text{Min} = 1 Calculate the range :   Range = 9 − 1 = 8 \text{Range} = 9 - 1 = 8
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Standard Deviation

Definition:        Standard deviation (SD) is a measure of the amount of variation or dispersion in a set of values. A low SD means the data points tend to be close to the mean, while a high SD indicates that the data points are spread out over a wider range. Formula: SD = ∑ i = 1 n ( x i − x ˉ ) 2 n \text{SD} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}} where x i x_i ​ represents each value, x ˉ \bar{x} is the mean, and n n is the number of values. Example: Given the dataset: [ 1 , 3 , 5 , 7 , 9 ] [1, 3, 5, 7, 9] Calculate the mean x ˉ \bar{x} : x ˉ = 1 + 3 + 5 + 7 + 9 5 = 25 5 = 5   \bar{x} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5 Calculate each squared deviation from the mean : ( 1 − 5 ) 2 = 16 , ( 3 − 5 ) 2 = 4 , ( 5 − 5 ) 2 = 0 , ( 7 − 5 ) 2 = 4 , ( 9 − 5 ) 2 = 16   (1-5)^2 = 16,\quad (3-5)^2 = 4,\quad (5-5)^2 = 0,\quad (7-5)^2 = 4,\quad (9-5)^2 = 16 Sum the squared deviations : ∑ ( x i − x ˉ ) 2 = 16 + 4 + 0 + 4 + 16 = 40   \s...
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Median

Definition:      The median is the middle value of a dataset when it is ordered from smallest to largest. If the dataset has an even number of observations, the median is the average of the two middle values. Steps to Calculate: Arrange the data in ascending order. If the number of observations (n) is odd, the median is the middle value. If n is even, the median is the average of the two middle values. Example: Given the dataset: [ 7 , 3 , 1 , 5 , 9 ] Arrange in ascending order: [ 1 , 3 , 5 , 7 , 9 ] The number of observations n = 5 (odd), so the median is the middle value: 5
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Mean (Average)

Definition: The mean is the average of a set of numbers. It's calculated by summing all the values and dividing by the number of values. Formula: Mean = ∑ i = 1 n x i n \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} where x i x_i ​ represents each value in the dataset, and n is the number of values. Example: Given the data: x = [ 2 , 4 , 6 , 8 , 10 ] x = [2, 4, 6, 8, 10] Mean = 2 + 4 + 6 + 8 + 10 5 = 30 5 = 6 \text{Mean} = \frac{2 + 4 + 6 + 8 + 10}{5} = \frac{30}{5} = 6
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