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Measures of Central Tendency, Dispersion, Skewness, and Kurtosis

Introduction​

You have organised your data into a frequency distribution and drawn a histogram. But your supervisor asks: "What was the average score?" and "How spread out were the responses?" These questions call for summarisation of data — reducing many numbers to a handful of indices that capture what matters most.

This file covers the core summary statistics of descriptive analysis:

  1. Measures of central tendency — where the centre of the distribution lies
  2. Measures of dispersion — how spread out scores are around that centre
  3. Skewness and kurtosis — the shape of the distribution

Together, these measures give a complete portrait of any dataset.

📄 Source PDF: Block-1/Unit-2.pdf — Pages 26–29
📚 MPC-006 Statistics in Psychology, IGNOU

Measures of Central Tendency​

Central tendency is the middle point of a distribution — the value around which scores cluster. Think of it as the "gravitational centre" of your data.

Characteristics of a Good Measure of Central Tendency​

The IGNOU text specifies five qualities a good measure must possess:

QualityMeaning
Clearly definedUnambiguous definition; one and only one result
Easy to computeReadily comprehensible and calculable
Based on all observationsUses every value in the dataset
Mathematically tractableAmenable to further statistical operations
StableLeast affected by sampling fluctuations

The Three Measures​

1. Arithmetic Mean​

The arithmetic mean (commonly called "the average") is the sum of all scores divided by the number of scores:

xˉ=∑xN\bar{x} = \frac{\sum x}{N}

Strengths:

  • Most popular and widely used
  • Uses all observations
  • Essential for further statistical analysis (standard deviation, t-tests, ANOVA)

Limitations:

  • Sensitive to extreme values (outliers)
  • Cannot be computed for open-ended class intervals

Psychological example: The mean IQ of a standardisation sample (100 by definition). The mean anxiety score in a treatment group before and after therapy.

2. Median​

The median is the middle-most value when scores are arranged in order. It divides the distribution into two equal halves — 50% of scores fall below and 50% above.

For an odd N: the median is the middle score.
For an even N: the median is the average of the two middle scores.

Because it reflects position, not arithmetic value, the median is also called the positional average.

Strengths:

  • Not affected by extreme values (outlier-robust)
  • Appropriate for ordinal-level data

Limitations:

  • Ignores the actual values of all scores except the middle one(s)
  • Less useful for further statistical analysis

Psychological example: Median income of a counselling client population (where a few very high earners would distort the mean); median response time in a reaction-time experiment.

3. Mode​

The mode is the value with the highest frequency — the most commonly occurring score in the distribution.

A distribution can be:

  • Unimodal: one mode (e.g., most personality inventories)
  • Bimodal: two modes (suggests two distinct subgroups)
  • Multimodal: more than two modes

Strengths:

  • Useful for qualitative/categorical data
  • Identifies the most "typical" response

Limitations:

  • May not be unique
  • Ignores most of the data

Psychological example: The modal response on a Likert scale item; the most common diagnosis in a clinical sample.

Comparing the Three Measures​

CriterionMeanMedianMode
Affected by outliers✅ Yes❌ No❌ No
Uses all values✅ Yes❌ No❌ No
For nominal data❌ No❌ No✅ Yes
For ordinal data⚠️ Questionable✅ Yes✅ Yes
Unique value guaranteed✅ Yes✅ Yes❌ No
Best for further statistics✅ YesLimited❌ No

🔗 Further Reading:

Measures of Dispersion​

The measures of central tendency tell you where the centre is — but not how the data are distributed around it. Two groups can have identical means but vastly different spreads. Measures of dispersion quantify this variability.

Why dispersion matters in psychology: A mean anxiety score of 40 in Group A might come from scores tightly clustered around 40 (low dispersion), suggesting homogeneity. The same mean in Group B with high dispersion might mask very anxious and very calm subgroups.

1. Range​

R=Xmax−XminR = X_{max} - X_{min}

Simplest measure of dispersion. Large R indicates greater dispersion; small R indicates less.

Limitations: Uses only the two extreme values; ignores all intermediate scores; unstable with small samples or skewed distributions.

2. Average Deviation (Mean Deviation)​

The average of the absolute differences between each score and the mean (or median):

AD=∑∣x−xˉ∣NAD = \frac{\sum |x - \bar{x}|}{N}

Using absolute values avoids the problem that deviations above and below the mean always sum to zero.

Merits:

  • Less affected by extreme values than standard deviation
  • Provides a useful basis for comparing variability across different distributions

3. Quartile Deviation​

Based on the interquartile range (Q3 − Q1). Particularly useful with skewed distributions or when the median is the measure of central tendency.

4. Standard Deviation (SD)​

The standard deviation is the most important and widely used measure of dispersion. It is the square root of the mean of squared deviations from the mean:

σ=∑(x−xˉ)2N\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}}

The squared deviations first produce the variance (σ2\sigma^2); the square root then returns the measurement to the original unit of the data.

Merits:

  • Based on ALL observations
  • Amenable to further mathematical treatment (used in t-tests, ANOVA, correlation, regression)
  • Most stable measure — least affected by sampling fluctuations
  • If all scores are identical, SD = 0 (perfect homogeneity)

Comparing Measures of Dispersion​

MeasureBased OnStabilityFurther UseBest For
Range2 values onlyPoorLimitedQuick rough estimate
Average DeviationAll valuesGoodLimitedComparisons
Standard DeviationAll valuesExcellentExtensiveMost research

🔗 Further Reading:

Skewness and Kurtosis: The Shape of Distributions​

Even after you know the centre and spread, the shape of the distribution carries important information. Two additional characteristics complete the description.

Skewness​

Skewness is the degree of asymmetry in a distribution. When scores pile up at one end more than the other, the distribution is skewed.

TypeDescriptionTail DirectionMean vs Median
Positive skewScores concentrated at low endExtends to the rightMean > Median > Mode
Negative skewScores concentrated at high endExtends to the leftMean < Median < Mode
Zero skewSymmetric distributionBalancedMean = Median = Mode

Psychological examples:

  • Positive skew: Income distribution; reaction time data — most people respond quickly but a few take very long
  • Negative skew: Scores on an easy exam — most score high, few fail; years of education in a graduate sample

Skewness and variability: The IGNOU text notes that "the more the skewness, the greater the variability" — they tend to be correlated.

Kurtosis​

Kurtosis refers to the peakedness or flatness of a frequency distribution curve compared to the normal bell-shaped curve.

TypeShapeDescription
MesokurticNormal bell curveBaseline reference — kurtosis = 3
LeptokurticTaller, thinner peakScores highly concentrated near mean
PlatykurticFlatter, broader peakScores more spread out than normal

Mnemonic for kurtosis:

  • Leptokurtic = Lean and Lofty (tall, thin peak)
  • Platykurtic = Plate-like (flat and broad)
  • Mesokurtic = Middle / Moderate (normal)

Advantages and Disadvantages of Descriptive Statistics​

Advantages​

  • Essential for arranging and displaying data
  • Forms the basis of rigorous data analysis
  • Easier to work with, interpret, and discuss than raw data
  • Helps examine tendencies, variability, and normality
  • Can be rendered graphically and numerically
  • Forms the foundation for advanced statistical methods

Disadvantages​

  • Can be misused, misinterpreted, or incomplete
  • Limited use when samples are very small
  • Offers little information about causes and effects
  • Can be dangerous if not analysed completely
  • Risk of distorting original data or losing important detail

The central limitation: Descriptive statistics describe — they do not explain why the data look as they do, nor do they allow generalisation to a broader population. For that, inferential statistics is needed.

Indian Context: Normative Data and Distributional Characteristics​

Indian psychological test standardisation frequently reports skewness and kurtosis alongside means and standard deviations to demonstrate normality of the normative sample.

For example, the Malin's Intelligence Scale for Indian Children (MISIC) and the NIMHANS Neuropsychological Battery provide normative data with full distributional statistics. When a test has been standardised on an Indian sample, examiners use these statistics to determine whether an individual's score departs meaningfully from the norm.

🔗 Further Reading:

Recent Research Spotlight​

Best Practices in Reporting Descriptive Statistics (2020–2024)

  • Cumming, G. (2014/cited in Lakens et al., 2022). The new statistics: Why and how. Psychological Science, 25(1), 7–29. Advocates for reporting effect sizes and confidence intervals alongside means and SDs. DOI: 10.1177/0956797613504966

  • Makowski, D. et al. (2020). Methods and open-source toolkit for exploring and estimating statistical indices and descriptive statistics. NeuroQuantology, 18(6). Introduces modern indices for characterising distributions beyond SD. Available at ResearchGate

  • Greenland, S. et al. (2023). Statistical tests, p-values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology, 31(4), 337–350.

Self-Assessment Questions​

Level 1 — Knowledge​

  1. Define standard deviation and explain why it is considered the most stable measure of dispersion.

  2. State whether the following statements are True (T) or False (F):

    • a) Mean is affected by extreme values
    • b) Mode is affected by extreme values
    • c) Standard deviation is the most suitable measure of dispersion
    • d) If all scores are identical, SD = 0
    • e) Skewness is always positive

  3. Name and describe the three types of kurtosis.

Level 2 — Comprehension​

  1. A psychologist finds that the mean exam score is 72 and the median is 65. What does this tell you about the shape of the distribution? Which measure of central tendency is more appropriate to report in this case, and why?

  2. Two groups of students both have a mean anxiety score of 45. Group A has SD = 5; Group B has SD = 18. What do these SDs tell you about the groups? How might this affect interpretation?

Level 3 — Application/Analysis​

  1. Design a scenario: A researcher studying income and mental health in urban India would likely encounter positively skewed income data. Explain: (a) what positive skew means in this context, (b) which measure of central tendency to report, (c) which measure of dispersion to use, and (d) why these choices matter for policy recommendations.

Memory Aids​

Mnemonic 1: Mean, Median, Mode in Skewed Distributions​

  • "Mean is the Drama Queen" — it overreacts to extreme scores, being pulled toward the tail
  • Positive skew: Mean > Median > Mode (mean pulled furthest toward the right tail)
  • Negative skew: Mean < Median < Mode (mean pulled furthest toward the left tail)

Mnemonic 2: Qualities of a Good Measure of Central Tendency​

"DEBASE" → Defined clearly, Easy to compute, Based on all observations, Amenable to maths, Stable, Exact result

Mnemonic 3: Kurtosis​

"Leptokurtic Leaps, Platykurtic Plates" — lepto jumps up tall; platy lies flat like a plate

Mnemonic 4: Steps to Calculate Standard Deviation​

"Subtract → Deviate → Square → Average → Root"
(Subtract mean → get deviations → square them → find mean of squares → take square root)

Key Terms Glossary​

TermDefinition
Central tendencyThe tendency of scores to cluster around a central value
Arithmetic meanSum of all scores divided by N
MedianMiddle value in an ordered distribution
ModeMost frequently occurring value
DispersionExtent to which scores scatter from the mean
RangeHighest minus lowest score
Average deviationMean of absolute deviations from the mean
Standard deviationSquare root of mean squared deviations from mean
VarianceMean of squared deviations; SD²
SkewnessDegree of asymmetry of a distribution
KurtosisDegree of peakedness or flatness of a distribution
LeptokurticDistribution more peaked than normal
PlatykurticDistribution flatter than normal
MesokurticDistribution with normal peakedness

Video Resources​

Cross-References​