Measure of Central Tendency

Topic 3: Measure of Central Tendency

  • Represents the center of a data set.

  • A single summary number indicates where many of the scores lie.

Types of Measures of Central Tendency

A. Mean

  • The arithmetic average of a data set.

  • Formula: Mean = Σx/n (sum of all scores divided by number of scores)

    • Example 1:

      • Scores: 1.42, 1.97, 1.42, 1.50, 1.67

      • Mean = 7.98 / 5 = 1.60

    • Example 2:

      • Scores: 2.00, 3.40, 7.00, 11.00, 23.00, 3.41

      • Calculation needed for Mean.

  • Can also be calculated using statistical software like Excel.

B. Median (mdn)

  • The midpoint of a data set.

  • Divides values into two equal parts, locating the middle value.

  • Calculation: Location of Median (L) = (n + 1) / 2

    • Example 1:

      • Scores: 11, 12, 13, 14, 15

      • L = (5 + 1) / 2 = 3 → Median = 13

    • Example 2:

      • Scores: 2, 4, 4, 6, 8, 8

      • L = (6 + 1) / 2 = 3.5 → Median = (4 + 6) / 2 = 5

C. Mode

  • The most frequently occurring value in a data set.

  • Example 1:

    • Scores: 1, 7, 5, 9, 8, 7 → Mode = 7

  • Example 2:

    • Scores: 1, 7, 5, 9, 8 → No Mode

  • Example 3:

    • Scores: 1, 7, 7, 5, 9, 9, 8 → Bimodal (Modes = 7 and 9)

  • Mode is primarily useful for categorical data or large datasets of interval/ratio data.

When to Use Each Measure

  1. Nominal Data: Use Mode (mean and median not applicable).

    • Example: Law = 64; Kine = 59; Eng. = 37

  2. Ordinal Data: Use Median.

    • Example: Positional rankings (1st, 2nd, 3rd, etc.).

  3. Interval or Ratio Data: Use Mean and/or Median.

    • Use Median if data is highly skewed or has outliers (as it's less affected).

    • Example of skew:

    • Player salaries → Mean = $495,000; Median = $125,000 due to outlier.

Excel: Calculating Central Tendency Measures

  • Excel Formulas:

    • Mean: = average(data_set)

    • Median: = median(data_set)

    • Mode: = mode(data_set)

  • Descriptive Statistics can be run using Toolpak in Excel to provide detailed statistics including mean, median, mode, range, standard deviation, and more.

Graphical Representation of Data

  • Summary data can be graphically depicted using:

    • Bar Graphs: Represent means or medians with categories on the x-axis.

    • Histograms: Display the frequency distribution of interval/ratio data.

Types of Data Distributions

  • Symmetrical Distribution: Frequencies decrease evenly on both sides of the center (e.g., mean = median = mode).

  • Skewed Distribution: Frequencies bunched at one end of the scale.

    • Positively Skewed: Bunched at lower scores; mean > median > mode.

    • Negatively Skewed: Bunched at higher scores; mode > median > mean.

Variability and Kurtosis

  • Spread: Refers to how much data varies.

  • Kurtosis: Indicates the peakness or flatness of a distribution.

    • Leptokurtic: More peaked, fewer scores in tails.

    • Platykurtic: Flatter distribution, more evenly spread out.

Importance of Graphing

  • Reliance solely on summary statistics can be misleading. Always visualize data through graphs to understand its distribution.