C4 Central Tendency

• The mode is the score that occurs with the greatest frequency.
• The median, Mdn, is the middle score when the observations are arranged in order of magnitude, so that an equal number of scores falls below and above.
• The arithmetic mean, X, is the sum of all scores divided by the number of scores.
• A symbol is needed to denote the operation of summation. This is found in the capital Greek letter sigma,Σ.
• The symbol for the number of observations: n.
• The mean is the balance point of a distribution.
• The mean is represented by the symbol X (“X-bar”).
• The mean is responsive to the exact position, or magnitude, of each score in the distribution.
• The sum of the deviations of scores from the mean always equals zero. That is, ΣX−X = 0.
• The grand mean is the mean of all N scores (just sum all scores and divide by the total sample size N )
• In a perfectly symmetrical distribution, one-half of the distribution is the mirror image of the other.
• In a normal distribution, as in Figure 4.4a, the mode shares the value of the mean and median.
• In a positively skewed distribution: mode < Mdn < X, the relative location of measures of central tendency (particularly the mean and median) may be used for making rough judgments about both the presence of skewness and its direction.
• In a negatively skewed distribution, X < median < mode, an equal proportion of scores falling above and below—it typically sits to the left of the mode.
• Three measures of central tendency are commonly encountered in the research literature: mode, median, and mean.
• The mode is the only appropriate measure of central tendency for qualitative, or nominal, variables.
• Because the mean has superior mathematical tractability and stability, it typically is the preferred measure of central tendency in statistical formulas and procedures.
• Sampling Stability: the mean would be the most stable of the three measures of central tendency—it would evidence the least“sampling variation.”
• Mathematical Tractability: when further statistical work is to be done, the mean will almost always be the most useful measure.
• Median = middle value of ordered data set
• Mean = sum of all observations / number of observations
• Stability: The mean is more stable than either the mode or median because it is less affected by extreme values (outliers).
• Mean = sum of all values divided by number of values
• Mode = most frequently occurring score
• Range = difference between highest and lowest scores
• Mode = most frequently occurring observation
• Range = difference between highest and lowest values
• Variance = average squared deviation from the mean
• The mean is sensitive to outliers.
• Range = difference between largest and smallest values
• Mode = most frequently occurring value(s)
• Range = difference between largest and smallest observation
• Standard Deviation = square root of variance
• Variance = average squared deviations from the mean