4  Measures of Central Tendency

Mode, median, mean, and when a typical value is not typical

Tip💻 SPSS Tutorial Available

Learn how to calculate measures of central tendency in SPSS! See the SPSS Tutorial: Calculating Measures of Central Tendency in the appendix for step-by-step instructions on using the Frequencies and Descriptives procedures, interpreting output, and reporting results in APA format.

4.1 Chapter roadmap

Central tendency is about choosing a single value to represent the “typical” outcome. That sounds simple until you work with real Movement Science data, where the meaning of “typical” depends on both the measurement scale and the shape of the distribution. Some variables are skewed (EMG amplitude, sway area), so a few large values can pull the mean toward the extremes[1,2]. Some variables are bounded (function scores), which can create ceiling effects that make averages hard to interpret. Some variables are ordinal (e.g., pain ratings, RPE), in which equal numerical steps do not necessarily reflect equal changes in the underlying experience. Other variables reflect mixtures of subgroups (training vs. control, different movement strategies, or responders vs. non-responders), where a single overall average would describe no one in the dataset[3]. In these situations, choosing the wrong “typical value” can hide the real pattern, distort comparisons, and lead to conclusions that sound precise but are not faithful to the data. Robust and resistant summaries (for example, the median or trimmed means) are common alternatives when outliers or skew are influential[4].

By the end of this chapter, you will be able to:

  • Define and interpret the mode, median, and mean.
  • Explain how the distribution shape affects each measure.
  • Recognize situations where the mean is misleading.
  • Choose a defensible measure of central tendency for Movement Science variables.
  • Explain log transformations and the geometric mean at a conceptual level.

4.2 Workflow used throughout Part II

Use this sequence whenever you summarize data:

  1. Identify the variable type and measurement scale.
  2. Visualize the distribution (at least once).
  3. Choose a measure of center that matches the distribution and the purpose.
  4. Pair center with spread (you will formalize spread in the next chapter).
  5. Write a one-sentence justification for your choice. 

4.3 Why we summarize

A single number can be helpful because it reduces complexity. In Movement Science, this is often the first step before comparing groups, evaluating change over time, or building prediction models. However, a summary is only useful if it represents something meaningful.

A key idea in this book is that “typical” depends on your purpose. If you are describing typical sprint performance, a mean may be appropriate when sprint times are roughly symmetric. If you are describing typical sway area or EMG amplitude, the median often better reflects the typical participant because these variables are often right-skewed.

NoteReal example: same study, different shapes

In the Core Dataset, sprint time often behaves differently than sway area or EMG amplitude.

  • Sprint times often appear more symmetric, so the mean may well describe a typical value.
  • Sway area and EMG amplitude are often right-skewed, so the median may better describe a typical participant.

The correct choice depends on the distribution and the question.

4.4 The mode

The mode is the most frequently occurring value, meaning it identifies the single value or category that appears most often in the dataset[2]. It is especially intuitive for categorical outcomes because it answers a simple question: which label appears most often? For numeric variables, the mode can be helpful when measurements naturally repeat (such as integer counts), but it can be unstable when values are recorded with high precision and most observations are unique.

  • For nominal variables, the mode is often meaningful (the most common category).
  • For discrete count variables, the mode can also be meaningful.
  • For continuous variables with many unique values, the mode may be unstable or uninformative.

In Movement Science, the mode is most useful when the variable is categorical (for example, group membership, injury status) or when the data have a small set of repeated numeric values. Some datasets are also multimodal, meaning they can have more than one mode. Multiple modes can occur when there are clear subgroups (for example, two common strategies that produce two clusters of values), when values are rounded or “heaped” (for example many observations at whole numbers), or when measurements come from mixed conditions that should have been separated.

4.4.1 When the mode is informative

  • The most common category in a sample (for example, the most common sport).
  • The most common pain rating is when people use the same few integers.
  • The most common count of balance errors.

4.4.2 When the mode is not informative

If sprint times are recorded to the thousandth of a second, most values are unique. In that case, the “most common value” may occur only once, which does not feel like a meaningful summary.

WarningCommon mistake

Reporting a mode for a continuous variable with many unique values can create the illusion of a “most common value” that does not exist in a practical sense.

4.5 The median

The median is the middle value when the data are ordered. Half the observations are below it, and half are above it. Another way to say this is that the median is the 50th percentile, meaning it marks the point where 50 percent of the sample lies at or below that value and 50 percent lies at or above it[2].

A major strength of the median is robustness. Because the median depends on rank order rather than the exact magnitude of every observation, a few extreme values can shift the mean substantially but have much less influence on the median[2,4]. This is especially useful in Movement Science outcomes that are commonly right-skewed (for example, EMG amplitude and sway area), or that occasionally include unusual trials (for example, a slip during a sprint). In those situations, the median often describes a typical participant more faithfully than the mean, and it supports an exploratory mindset where you summarize what the data are actually doing before imposing stronger modeling assumptions[3].

4.5.1 How to find the median with 10 observations

When the sample size is even (like 10 observations), there is no single middle value. The median is defined as the value halfway between the two central ordered observations.

Use this example dataset of 10 observations (unsorted):

\[ 12,\ 9,\ 15,\ 8,\ 10,\ 14,\ 7,\ 11,\ 13,\ 9 \]

First, sort the values:

\[ 7,\ 8,\ 9,\ 9,\ 10,\ 11,\ 12,\ 13,\ 14,\ 15 \]

Method 1: Middle two and average (most common)

  • With 10 values, the middle two are the 5th and 6th values in the sorted list.
  • The 5th value is 10 and the 6th value is 11.
  • Median = ((10 + 11)/2 = 10.5).

Method 2: Position rule ((n+1)/2) (equivalent)

  • Compute the median position: ((n+1)/2 = (10+1)/2 = 5.5).
  • A position of 5.5 means the median is halfway between the 5th and 6th ordered values.
  • So the median is the average of 10 and 11, which is 10.5.

4.5.2 When the median is a strong choice

  • Right-skewed variables such as sway area and EMG amplitude.
  • Ordinal scales such as pain ratings and RPE.
  • Outcomes with occasional extreme values (for example, rare, very slow sprint times due to a slip).

4.5.3 Median as “typical participant”

For skewed distributions, the median often matches the value that feels typical when you look at the data. In that sense, it can be a better representation of a typical person’s outcome than the mean.

NoteReal example: ordinal summaries

For pain ratings (0–10) and RPE (6–20), the median is often more interpretable than the mean because the scales are ordinal and people cluster around a few values.

A common reporting pair is:

  • median with an interquartile range (IQR)

You will study IQR in the variability chapter.

4.6 The mean

The mean is the arithmetic average, computed by adding all observed values and dividing by the number of observations[2]. Conceptually, you can think of the mean as the “balance point” of the distribution: if each observation were a weight placed on a number line, the mean is where the distribution would balance[2]. This balance-point property helps explain both why the mean is useful and why it can mislead. Because every value contributes to the sum, extreme values and long tails pull the mean toward them, sometimes producing a “typical” value that few participants actually show[1]. In many Movement Science variables that are approximately symmetric, the mean is an efficient and widely used summary, but when skew or outliers are influential, you should visualize and consider resistant alternatives (such as the median or trimmed means) before treating the mean as typical[4].

\[ \bar{x} = \frac{\sum x}{n} \]

The mean is widely used because it has useful mathematical properties that make later methods work smoothly. Many inferential procedures are built around means and mean differences because the mean naturally connects with ideas such as sampling distributions, standard errors, confidence intervals, and hypothesis tests in common models[2]. At the same time, the mean uses every value in the dataset, so it is sensitive to extreme observations and to long tails. This sensitivity is not a flaw when the distribution is roughly symmetric, but it becomes a practical problem when a small number of unusually large or small values pull the mean away from where most observations lie[1]. For that reason, the mean should be treated as a default only after you have inspected the distribution and judged whether outliers or skew are influential, and you should be ready to report resistant summaries (such as the median) when they better represent a typical participant[4].

4.6.1 How to find the mean with 10 observations

Use this example dataset of 10 observations (unsorted):

\[ 12,\ 9,\ 15,\ 8,\ 10,\ 14,\ 7,\ 11,\ 13,\ 9 \]

Step 1: Add the values

\[ 12 + 9 + 15 + 8 + 10 + 14 + 7 + 11 + 13 + 9 = 108 \]

Step 2: Divide by the number of observations

  • Here, (n = 10)

\[ \bar{x} = \frac{108}{10} = 10.8 \]

Interpretation

  • The mean is 10.8, which does not need to be a value in the dataset.
  • If this variable were sprint time (seconds), a mean of 10.8 seconds could represent typical performance when the distribution is roughly symmetric.

A quick sensitivity check (why the mean can shift)

If a single observation is unusually large, the mean can move noticeably. For example, if the value 15 were replaced by 25 (an extreme performance or a recording error), the new sum would be 118 and the new mean would be:

\[ \bar{x}_{new} = \frac{118}{10} = 11.8 \]

That is a full 1.0 unit increase from changing only one of the ten values. This illustrates why the mean should be paired with a visualization and why resistant summaries (such as the median) can be more stable when outliers are influential[1,2].

4.6.2 When the mean is appropriate

  • Quantitative variables that are roughly symmetric and unimodal.
  • Situations where extreme values are rare and not overly influential.
  • When you need compatibility with methods built around means (for example, certain ANOVA models), as long as the assumptions are reasonable.

4.6.3 Mean and sample size

With small samples, the mean can bounce around from sample to sample because a single unusual value has a large influence. This does not mean you should never report a mean, but it does mean you should visualize and report uncertainty.

WarningCommon mistake

Reporting the mean for a heavily skewed variable without showing a distribution plot can make results look cleaner and more stable than they really are.

4.7 How distribution shape affects mean, median, and mode

When a distribution is roughly symmetric and unimodal, the mean, median, and mode tend to be similar. When the distribution is skewed, the tail pulls the mean toward the tail.

Distribution shape Typical ordering Why it happens
Symmetric, unimodal mean ≈ median ≈ mode no long tail pulling the mean
Right-skewed mean > median > mode high tail pulls mean upward
Left-skewed mean < median < mode low tail pulls mean downward

This ordering is a quick diagnostic, not proof. You should still visualize the distribution.

4.8 When the mean lies to you

The mean “lies” when it suggests a typical value that does not represent most observations. The number is still mathematically correct, but the story it tells can be misleading if you treat it as the typical participant. This is why experienced analysts routinely pair means with distribution plots and why resistant summaries (like the median) are often preferred when skew or outliers are influential[2,4].

A helpful way to think about it is this: the mean answers “what is the balance point of the data?” not necessarily “what value do most people have?”[2]. Those can be the same in a roughly symmetric distribution, but they can diverge sharply in the situations below.

4.8.1 Outliers

One extreme value can shift the mean noticeably, especially with small samples, because that single value contributes directly to the total sum. In performance testing, an unusual value might reflect a true event (slip, missed start, distraction), a protocol issue (inconsistent warm-up), or a recording problem (timing gate error). Either way, the mean may no longer represent typical performance.

What to do in practice:

  • First, verify whether the outlier is plausible (check notes, units, and the validity of the trial).
  • Second, visualize the distribution with the outlier present.
  • Third, report a resistant summary (e.g., median) alongside the mean, or justify using a trimmed mean if your goal is still a mean-like summary[4].

4.8.2 Skewed distributions

For strongly skewed variables, the mean can exceed what most participants show. In right-skewed variables like sway area and EMG amplitude, a small number of high values can pull the mean upward, producing a “typical” value that sits above the bulk of the data[1,2].

A quick check is to compare the mean and the median. If the mean is notably higher than the median and the histogram shows a long right tail, the mean is probably describing the tail as much as it is describing typical values[1]. In that case, the median (or geometric mean after a log transform) often communicates “typical” more faithfully.

4.8.3 Bounded scales and ceiling effects

Bounded scales, such as function scores (0–100), can produce ceiling effects. If many participants score near 100, the distribution piles up at the top, and the mean becomes hard to interpret as typical. Two different situations can produce the same mean:

  • many participants clustered near 100, with a few lower values
  • a wide spread of values across the full range

A plot is the only fast way to see which situation you have. Also note that ceiling effects can make change look smaller than it is because participants have limited room to improve. In those cases, medians and distribution plots often communicate more than means alone.

4.8.4 Mixtures of subgroups

If your data come from two different subgroups, the overall mean can fall between them and describe nobody. This happens when you combine groups that differ (e.g., training vs. control), or when the sample contains two distinct performance strategies that produce two clusters.

A simple numeric illustration:

  • Group A mean sprint time = 3.20 s
  • Group B mean sprint time = 3.80 s
  • Combined mean = 3.50 s

A combined mean of 3.50 s may be a value that few, if any, participants actually demonstrate. In this setting, the meaningful summaries are usually group-specific (means or medians within each group) plus a visualization that shows separation or overlap between subgroups[3].

4.8.5 Averaging across time points

Averaging pre, mid, and post values into one mean can hide the very phenomenon you are studying. If the scientific question is adaptation over time, collapsing across time points turns a trajectory into a single number. Two participants could have the same overall mean but very different patterns (steady improvement vs. improvement followed by regression). For repeated-measures designs, it is usually more informative to summarize by time point and show change with paired or trajectory plots.

NoteReal example: the grand mean problem

Imagine training participants improve sprint time from pre to post, while control participants stay the same. If you average all time points and both groups together, you create a mean that does not describe any specific group at any specific time. The mean is numerically correct but scientifically unhelpful. A more interpretable approach is to report group-specific summaries at each time point and visualize change over time before moving to inferential comparisons[2,3].

4.9 Optional tools: trimmed means and winsorized means

Sometimes the mean is useful, but a few extreme values are overly influential. Two related ideas can reduce the influence of outliers.

  • A trimmed mean removes a small percentage of the lowest and highest values, then averages the remaining values.
  • A winsorized mean replaces extreme values with less extreme boundary values, then averages them.

These approaches can help when outliers are mild, and you want a mean-like summary. They do not solve strong skew or mixtures of subgroups. In those cases, changing the summary measure or transforming the variable is usually more defensible.

4.10 Log transformations and the geometric mean

Some variables are better described on a multiplicative scale than an additive one. EMG amplitude and sway area are common examples because values are strictly positive and often show right-skew with occasional very large observations. In these settings, equal ratios can be more meaningful than equal differences. For instance, an increase from 20 to 40 units (a doubling) is often more comparable to an increase from 50 to 100 units (also a doubling) than it is to an increase of “+20 units” in every case. When the scientific story is about proportional change, measurement error is proportional to the size of the value, or the distribution looks approximately log-normal, a logarithmic transformation can reduce skew, stabilize variability, and make mean-based summaries and models behave more reasonably[5,6].

A quick Movement Science example: suppose sway area values include 8, 9, 10, 11, 40 cm². The arithmetic mean is pulled upward by the 40. On the log scale, the gap between 11 and 40 is compressed, so the “typical” value on the transformed scale is less dominated by the tail. Back-transforming the mean on the log scale yields a geometric mean, which often tracks what you would call a typical participant in this kind of right-skewed data[6].

4.10.1 What a log transform does conceptually

A log transform compresses large values more than small values. That compression reduces the influence of a long right tail and can make the distribution easier to summarize with mean-based methods[5].

Two practical consequences matter most in Movement Science:

  • Values that differ by the same ratio are equally spaced on the log scale. A doubling (for example, 20 → 40) and another doubling (50 → 100) represent the same log change.
  • Differences on the log scale correspond to multiplicative differences on the raw scale. This makes it natural to describe changes as percent-like or ratio-like effects rather than as constant additive shifts[5].

A brief numeric example (showing compression):

  • Raw EMG RMS (µV): 20, 30, 120
  • Differences on the raw scale: 30 − 20 = 10, but 120 − 30 = 90

On a log scale, the second jump is not nine times “bigger” than the first; it is brought closer, which often better matches how variability behaves in these measures.

Important detail: the choice of log base (log10 vs natural log) changes the scale but not the substantive conclusions. Base choice mainly affects the numeric values you see, not the pattern you interpret.

A useful interpretation shift is:

  • differences on the raw scale become ratios on the log scale
  • changes can be described in percent-like terms

4.10.2 The geometric mean

The geometric mean is closely linked to the log transform. One way to define it is:

  • Take the log of each value
  • Compute the mean on the log scale
  • Transform back to the original scale

Mathematically, for positive values (x_1,,x_n), the geometric mean is:

\[ \text{GM} = \left(\prod_{i=1}^{n} x_i\right)^{1/n} = \exp\left(\frac{1}{n}\sum_{i=1}^{n} \ln(x_i)\right) \]

The result is the geometric mean. It often represents a more meaningful typical value for right-skewed, approximately log-normal data because it reflects a “central” ratio rather than a “central” difference[5,6].

A short example:

  • Values: 10, 20, 40
  • Arithmetic mean: ((10+20+40)/3 = 23.3)
  • Geometric mean: ((10)^{1/3} = (8000)^{1/3} = 20)

Here, 20 is often closer to what you would call a typical value because the data reflect multiplicative spacing (each step is a doubling). In Movement Science, this logic frequently fits variables such as EMG amplitude, sway area, concentrations, and other positive measures where variability tends to scale with magnitude[6].

WarningCommon mistake

You cannot take the log of zero or negative values. If a variable can be zero (for example, an error count, a symptom count, or a variable that can legitimately be “none”), you need a plan before using logs[5].

Examples of reasonable plans:

  • Use a different summary that does not require transformation (for example, report the median and IQR for a right-skewed count).
  • Use a model designed for counts rather than forcing a normal-based approach.
  • If you add a small constant (for example \((\log(x+1))\)), state exactly what you added and why, and check whether conclusions are sensitive to that choice.

Avoid treating a constant-addition trick as automatic. If the dataset contains many zeros, the scientific meaning of “multiplicative change” may not match the structure of the outcome, and a count-focused approach is often more defensible.

4.11 Reporting central tendency responsibly

A strong summary communicates both the typical value and enough context to interpret it.

Practical reporting patterns:

  • Categorical variables: counts and percentages
  • Roughly symmetric quantitative variables: mean with a variability measure
  • Skewed quantitative variables: median with a variability measure
  • Log-normal style variables: geometric mean (and a variability measure appropriate for that context)

Always include units and specify the group and time point when relevant.

TipCentral tendency decision checklist

Before you report a center measure, confirm:

  • The variable type and scale make the summary meaningful.
  • You have visualized the distribution at least once.
  • The chosen center represents a typical value for most participants.
  • You did not average across groups or time points in a way that hides the question.
  • You paired the center with the spread and included units.

4.12 Worked example using the Core Dataset

This worked example emphasizes decision-making rather than computation. The key skill is choosing a summary that matches the variable.

4.12.1 Sprint time

Sprint time is typically summarized with a mean when the distribution is roughly symmetric. If a few extreme times occur (for example, a slip), the median may be a better representation of typical performance, or you may justify excluding invalid trials with documentation.

One-sentence justification example: The sprint time distribution is roughly symmetric at pre, so the mean summarizes typical performance, and we will pair it with a variability measure.

4.12.2 Sway area or EMG amplitude

Sway area and EMG amplitude are often right-skewed. A median often better represents the typical participant. If you plan to use mean-based modeling, a log transform and geometric mean may be appropriate.

One-sentence justification example: Sway area is right-skewed, so the median summarizes typical values more faithfully than the mean.

4.12.3 Pain rating or RPE

Pain ratings and RPE are ordinal scales. The median is usually more interpretable than the mean because the difference between adjacent numbers may not represent equal perceived differences.

One-sentence justification example: Pain rating is ordinal and clustered, so the median is the most defensible measure of center.

4.12.4 Function score

Function scores are bounded and can have ceiling effects. If many participants score near the maximum, the mean may hide the ceiling. The median can be useful, and visualization is essential.

One-sentence justification example: Function scores show a ceiling cluster, so the median better represents typical performance and highlights the boundary effect.

4.13 Chapter summary

Central tendency describes a “typical” value, but what counts as typical depends on the variable’s measurement scale, the distribution’s shape, and the scientific purpose[2,3]. The mode is most useful for categorical variables and can be informative when data are multimodal, which may signal subgroups, rounding, or mixed conditions. The median is the 50th percentile and is resistant to extreme values because it depends on rank order rather than the magnitude of every observation, making it especially defensible for skewed variables and ordinal scales such as pain ratings and RPE[2,4]. The mean is the balance point of the distribution and underlies many inferential methods, but it can be pulled by outliers, long tails, subgroup mixtures, or inappropriate collapsing across groups or time points, so it should be treated as a default only after you have inspected the data[1,2]. When the mean is misleading, you can report resistant alternatives (median, trimmed or winsorized means) or change the scale of analysis. For strictly positive, right-skewed outcomes where proportional differences are meaningful, log transformations can reduce skew and stabilize variability, and the geometric mean often provides a more representative typical value on the original scale[5,6].

4.14 Key terms

central tendency; mode; median; mean; skew; outlier; ceiling effect; grand mean; trimmed mean; winsorized mean; log transformation; geometric mean; log-normal

4.15 Practice: quick checks

The median is usually the most defensible first choice because it is resistant to the influence of a small number of very large sway-area values. In a right-skewed distribution, the mean is pulled upward toward the tail and can end up higher than what most participants show. If you need a mean-like summary for modeling, a log transform and geometric mean can also be appropriate for strictly positive, approximately log-normal sway measures.

The mean increases because the extreme time directly increases the sum of all sprint times, and with only 12 participants that single value can have a noticeable impact. The median changes little (or not at all) because it depends on rank order rather than the magnitude of the extreme value. In practice, you would first verify whether the slipped trial should be treated as invalid (based on protocol rules and notes) and then summarize with both a visualization and a resistant center measure if needed.

A ceiling effect can cause the mean to hide the distribution’s pile-up near the upper bound. Two datasets can share the same mean even if one is tightly clustered at 95–100 and another is widely spread. Ceiling effects also limit observable improvement, so mean changes can look small even when participants are meaningfully different at baseline. A distribution plot and summaries like the median and IQR often communicate this structure more clearly.

You may have hidden group differences, time trends, and group-by-time differences. Collapsing across groups can create a grand mean that describes neither group, and collapsing across time can erase adaptation patterns. Two participants or two groups can have the same overall mean while showing very different trajectories (improve then plateau, improve then regress, no change). A more informative approach is to summarize by group and time point and visualize change before inferential comparisons.

EMG amplitude is strictly positive and often right-skewed, with variability that scales with magnitude. In that setting, proportional changes (ratios) are often more meaningful than additive changes. The geometric mean reflects a central tendency on a multiplicative scale, so it is less dominated by extreme high values and can better represent a typical participant when the data are approximately log-normal.

NoteRead further

Look for short introductions to robust statistics and log-normal data in applied physiology or biomechanics methods texts. The key goal is not advanced math. The goal is to choose summaries that reflect the data’s structure and the scientific question.

TipNext chapter

The next chapter focuses on variability, including range, variance, standard deviation, and the interquartile range. You will learn why a measure of center without a measure of spread is incomplete.