Statistical Measures: Mode, Median, Mean, Range, and Standard Deviation

Overview of Measures of Central Tendency

  • Mode: The mode is the most frequently occurring value in a dataset.

    • Example: In a dataset of heights, if 154 appears three times, then the mode = 154.

  • Median: The median is the middle value when the dataset is ordered.

    • To find the median, arrange data from smallest to largest.

    • Example: For a dataset with values arranged, if 154 is in the middle, then median = 154.

    • For odd n: Median position formula = ( \frac{n+1}{2} )

    • Example: For n = 9, median position = ( \frac{9+1}{2} = 5 ).

    • For even n: Average the two middle values.

    • Example: For n = 10, if middle values are 154 and 155, median = ( \frac{154 + 155}{2} = 154.5 ).

  • Mean: The arithmetic average of the dataset.

    • Mean formula: ( \bar{x} = \frac{\sum x_i}{n} )

    • Example: If total sum of values = 1656 and n = 10, then ( \bar{x} = \frac{1656}{10} = 165.6 ).

Differences Between Mean and Median

  • Median: Represents the middle point (physical center).

  • Mean: Represents the balance point or average.

Overview of Measures of Spread

  • Range: Difference between the maximum and minimum values.

    • Range formula: ( \text{Range} = \text{max} - \text{min} )

    • Example: For max = 196 and min = 139, Range = ( 196 - 139 = 57 ).

  • Standard Deviation: Measures how spread out the values are in a dataset.

    • Standard deviation formula: ( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} )

    • Steps to calculate:

    1. Calculate the mean ( \bar{x} ).

    2. Subtract the mean from each data value, square it, and sum the results.

    3. Divide by n, and take the square root to find the standard deviation.

    • Example: If the numerator (sum of squares) is 75.2 for n = 5, then ( s = \sqrt{\frac{75.2}{5}} \approx 4.336 ).

Interpretation of Measures of Spread

  • Standard Deviation:

    • Small standard deviation: values are closely clustered around the mean, indicating low variability.

    • High standard deviation: values are more spread out from the mean, indicating high variability.

  • Variance: Related to standard deviation, computed similarly without taking the square root.

    • Notation: Variance is denoted as ( s^2 ) and standard deviation as ( s ).

    • Variance explains the extent of distribution variance among data points.