Unit 3: Measures of Central Tendency and variability

I. Orientation

Statistical description begins by reducing a set of psychological observations to representative values and measures of spread. Measures of central tendency identify a typical or central score, while measures of variability show how consistently or widely scores differ from that centre. In psychological research, these measures are applied to variables such as reaction time, intelligence scores, symptom ratings, memory scores, and questionnaire responses.

• Data arrangement: Scores may be raw, ordered, grouped into frequency distributions, or divided into intervals.
• Central tendency: The mean, median, and mode describe the location of a distribution.
• Variability: The range, interquartile range, average deviation, variance, and standard deviation describe dispersion.
• Notation: (X) represents an individual score; (N) represents the number of scores in a population; (n) represents the number of scores in a sample; (f) represents frequency.
• Units: Central-tendency measures generally retain the original unit, while variance is expressed in squared units; standard deviation returns to the original unit.
• Distributional sensitivity: The mean and standard deviation are strongly influenced by extreme scores, whereas the median and interquartile range are more resistant.

II. The Median — the positional centre

A. the median

The median is the middle value when scores are arranged from smallest to largest or largest to smallest.

• Odd number of scores: For (N) ordered observations, the median is at position ((N+1)/2).
• Even number of scores: The median is the average of the values at positions (N/2) and ((N/2)+1).
• Worked example: For (3, 5, 7, 9, 12), the third score is the median: (7). For (3, 5, 7, 9), it is ((5+7)/2=6).
• Grouped data: The median class contains the observation at the (N/2) position. An estimated grouped-data median is:

Median = L + [(N/2 − CF) / f] × h

• Symbol meanings: (L) is the lower boundary of the median class; (CF) is cumulative frequency before that class; (f) is its frequency; (h) is class width.
• Strength: It is suitable for skewed data, ordinal data, and distributions containing extreme scores.
• Limitation: It does not use the exact distance of every score from the centre, so it is less mathematically useful than the mean for many inferential procedures.

III. The Mode — the most frequent value

A. the mode

The mode is the score or category occurring most frequently in a distribution.

• Raw scores: In (2, 3, 3, 4, 5, 5, 5, 7), the mode is (5), because it occurs three times.
• Frequency interpretation: The mode is identified by the highest frequency, not by its position or numerical size.
• Multiple modes: A distribution may be bimodal or multimodal if two or more values share the greatest frequency.
• Categorical data: The mode is the only central-tendency measure directly applicable to nominal categories, such as the most common therapy preference.
• Grouped data: The modal class is the interval with the highest frequency; its exact value may be estimated from neighbouring frequencies.
• Strength: It is simple, unaffected by extreme scores, and useful for discrete or nominal observations.
• Limitation: A distribution may have no unique mode, and a small change in frequencies can change the mode substantially.

IV. The Mean — the arithmetic balance point

A. the mean

The arithmetic mean is obtained by adding all scores and dividing by the number of scores.

Population mean: μ = ΣX / N
Sample mean:     X̄ = ΣX / n

• Symbol meanings: (\mu) is the population mean; (\bar X) is the sample mean; (\Sigma X) is the sum of scores; (N) and (n) are population and sample sizes.
• Worked example: For (4, 6, 8, 10), (\bar X=(4+6+8+10)/4=7).
• Balance property: Deviations from the mean sum to zero:

Σ(X − X̄) = 0

• Grouped data: The mean can be calculated as (\bar X=\Sigma fX/\Sigma f), where (f) is frequency and (X) is the score or class midpoint.
• Strength: It uses every observation and supports algebraic operations, correlation, regression, variance, and standard deviation.
• Limitation: It is sensitive to extreme scores. A very long reaction time or unusually high symptom score can pull the mean toward itself.
• Distributional relationship: In a symmetric unimodal distribution, mean, median, and mode are approximately equal; in a positively skewed distribution, the mean usually exceeds the median.

V. Range — the simplest spread measure

A. range

The range is the difference between the largest and smallest observations.

Range = Xmax − Xmin

• Symbol meanings: (X{\max}) is the largest score and (X{\min}) is the smallest score.
• Worked example: If reaction times range from (310) ms to (590) ms, the range is (590-310=280) ms.
• Interpretation: A larger range indicates greater overall separation between the two extreme observations.
• Strength: It is quick to calculate and easy to communicate.
• Limitation: It depends only on two scores and is therefore unstable. Adding one unusually high score can increase the range greatly without changing most observations.
• Use: It is useful for an initial description, but it should normally be accompanied by a more stable measure such as the interquartile range or standard deviation.

VI. Interquartile and the semi-interquartile range — central fifty percent

A. interquartile and the semi-interquartile range

The interquartile range measures the spread of the middle 50% of ordered scores, while the semi-interquartile range gives half of that spread.

• Quartiles: (Q_1) is the 25th percentile and (Q_3) is the 75th percentile. They divide ordered data into four equal parts.
• Interquartile range:

IQR = Q3 − Q1

• Semi-interquartile range:

SIQR = (Q3 − Q1) / 2

• Worked example: If (Q_1=42) and (Q_3=66), then (IQR=24) and (SIQR=12) score units.
• Interpretation: The IQR describes the width of the central half; the SIQR describes the typical distance from the median to either quartile when the distribution is reasonably balanced.
• Strength: Both measures resist the influence of extreme scores and are appropriate with the median for skewed distributions.
• Limitation: They ignore the lowest 25% and highest 25% of observations, so they do not use all available information.
• Relationship to the median: The median is the 50th percentile and lies between (Q_1) and (Q_3).

VII. The Average Deviation — mean absolute distance

A. the average deviation

Average deviation, also called mean deviation or mean absolute deviation, is the average of the absolute distances of scores from a selected central value.

Average deviation about the mean = Σ|X − X̄| / N
Average deviation about the median = Σ|X − Md| / N

• Symbol meanings: (|\ |) indicates absolute value; (X) is an observation; (\bar X) is the mean; (Md) is the median.
• Why absolute values are used: Without absolute values, positive and negative deviations would cancel, producing zero around the mean.
• Worked example: For (2, 4, 6), the mean is (4); deviations are (-2,0,+2), absolute deviations are (2,0,2), and average deviation is (4/3=1.33).
• Grouped data: Use (\Sigma f|X-A|/\Sigma f), where (A) is the chosen centre and (f) is frequency.
• Interpretation: A value of (1.33) means that scores are, on average, 1.33 units from the selected centre.
• Strength: It is expressed in original units and is easier to interpret than variance.
• Limitation: Absolute values make algebraic manipulation more difficult; consequently, standard deviation is more common in advanced statistical analysis.

VIII. The Standard Deviation — typical distance from the mean

A. the standard deviation

Standard deviation is the square root of the average squared deviation from the mean. It is the most widely used measure of variability for interval- and ratio-level psychological data.

Population SD: σ = √[Σ(X − μ)² / N]

Sample SD:     s = √[Σ(X − X̄)² / (n − 1)]

• Symbol meanings: (\sigma) is population standard deviation; (s) is sample standard deviation; (\mu) and (\bar X) are population and sample means.
• Why deviations are squared: Squaring removes negative signs and gives greater weight to large departures from the mean.
• Why samples use (n-1): The (n-1) denominator, called degrees of freedom, corrects the tendency of a sample to underestimate population variability.
• Worked example: For population scores (2,4,6), (\mu=4); squared deviations are (4,0,4), so (\sigma=\sqrt{8/3}\approx1.63).
• Interpretation: A smaller standard deviation indicates scores cluster more closely around the mean; a larger value indicates greater spread.
• Normal-distribution rule: In an approximately normal distribution, about 68% of scores lie within (1) SD, about 95% within (2) SDs, and about 99.7% within (3) SDs of the mean.
• Limitation: It is sensitive to outliers and is less representative when the distribution is highly skewed.

IX. Variance — squared dispersion

A. the variance

Variance is the average squared deviation from the mean and is the mathematical foundation of standard deviation and many inferential tests.

Population variance: σ² = Σ(X − μ)² / N
Sample variance:     s² = Σ(X − X̄)² / (n − 1)

• Symbol meanings: (\sigma^2) and (s^2) are population and sample variances; all other symbols have the meanings defined above.
• Connection with standard deviation:

σ = √σ²       and       s = √s²

• Units: If scores are measured in points, variance is measured in points squared; taking the square root restores points.
• Worked example: For (2,4,6), the population variance is (8/3\approx2.67), and the population standard deviation is (\sqrt{2.67}\approx1.63).
• Strength: Variance is essential in analysis of variance, regression, reliability analysis, and the decomposition of total variability.
• Limitation: Its squared units make direct interpretation less intuitive than standard deviation.
• Analytical meaning: Variance quantifies how much individual observations contribute to total dispersion through their squared deviations from the mean.