# RELIABILITY

Reliability helps us understand the sources of error and how they affect findings in practice and in research.

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By **Dr. Jérôme Sauret**

3rd April 18 | 10 min read

**Table of Contents**

- Summary
- Introduction
- What is Reliability?
- The 3 Types of Reliability
- Why is Reliability important?
- How to determine/measure Reliability?
- Future Research and Recommendations
- Conclusion
- References
- About the Author
- Comments

Precise measurements are central to sports science practice and research; however, error is an inherent part of testing. Reliability helps us understand the sources of error and how they affect findings in practice and in research. This article defines reliability and provides some key points for a better understanding of the statistical and practical concepts.

**Keywords**: reliability, correlation, coefficient of variation, limits of agreements

In Sports Science, testing is a central tool to support decisions made by practitioners and researchers alike. Scoring (X), for example, vertical jump height, is comprised of the true score (T) and the error (e), and it can be expressed in the following equation [1]:

X = T + *e*

Error is thought to originate in three main sources [2]:

**The participant**: physiological and/or mental factors. For example, the stress of upcoming exams may decrease motivation and strength.

**The tester**: change of technique or of judgment. For example, slightly modifying the way they use the skinfold callipers to measure body fat, resulting in biased readings.

**Technical error**: due to the device used or the setup/environment. For example, if you step on a set of scales twice, only a few seconds apart, the readings will be different due to equipment measurement error, not weight change.

Error decreases the precision of a test and increases the smallest detectable change, making research and applied sport science less sensitive. For example, if a strength and conditioning coach monitors strength and finds a 5kg increase in back squat 1RM, then:

- With a precision of ±2 kg they can confidently say that performance improved.
- With a precision of ±5 kg they cannot say that performance improved.

Error also impairs our ability to make predictions or to classify individuals, which may be a problem depending on the outcome and population. If a nutritionist measures body fat with 5% error (e.g. 26% instead of 21%), this could have profound effects on how the subject is classified. For example:

- Healthy professional football player: this may not be a big problem.
- An obese patient trying to lose weight.

To minimise error and improve clinical and research practices, standardisation and documentation of the following is critical:

**Test preparation**: e.g. Familiarisation procedure and test timing.**Testers training:**g. Verification of Inter-tester reliability with the gold standard.**Testing procedures**: e.g. Implementation of an adequate and standardised warm-up.

Reliability is the study of error or score variance over two or more testing occasions [3], it estimates the extent to which the change in measured score is due to a change in true score. Theoretically, a perfectly reliable measure would produce the same score over and over again, assuming that no change in the measured outcome is taking place.

In the above schematic we can see:

- X1 and X2: The two repeated measurements on the same individual for the test (X)
- e1 and e2 : The random errors for measurements 1 and 2, respectively.
- T: The true value of the test

This shows that practice is much more complex and that partitioning error and score variance may be challenging for the practitioner and the researcher alike.

Different terminologies are used and we will briefly differentiate them [4]:

**Repeatability**is the variation in measurement for the same subjects under the same conditions over a short period of time [3] (e.g. the difference between two weightings, only a few minutes apart, on the same scale).

**Reproducibility**is the variation in measurement for the same subjects under different conditions over a short period of time (e.g. the difference between two weightings, only a few minutes apart, on two different scales (inter-rater and methods comparisons).

It is important to understand that there are three types of reliability [5], all of which are discussed below.

One of the first things to learn when attempting to understand reliability is to know that there are 3 different types of reliability: 1) Change in Mean; 2) Correlation; and 3) Within-Participant Variation.

**1. Change in the Mean: Bias**

As indicated in the name, it is the group mean change over two or more testing occasions and it has two components:

**Random error**of measurement (e1 and e2 in the above diagram).**Systematic change**: change in the true value (T in the above diagram).

**2. Correlation or Relative Reliability**

Intraclass Correlation Coefficient (ICC) is considered as the most relevant indicator of relative reliability [2]. It measures the linearity of the relationship between two repeated measures and represents how well the rank order of participants in one trial is replicated in a second trial (e.g. runners performing a 5k twice and finishing with the same ranking).

Figure 1 illustrates a variety of possible scenarios when repeated measures are plotted against each other. A straight and narrow line indicates the correlation is close to 1 or -1, whilst a cloud of data points demonstrates no relation (i.e. ICC closer to 0).

**3. Within-Participant Variation: Absolute Reliability**

Absolute reliability looks into the random error between repeated measures (e1 and e2) for one participant. Since it is expressed in the original unit (e.g. kg or cm), or in a proportion of it, it is useful to gauge the precision of a measure [2, 3]. It is, therefore, most relevant to assess the reliability of measures that evaluate change (e.g. tracking strength) [2, 3].

Next, we will briefly present the most common methods to investigate the within-participant variation:

**Standard Error of Measurement (SEM)**

The SEM is an estimate of the absolute value of the typical deviation between the observed scores and the true score, which is assumed to be the mean of all measured values [6].

**Coefficient of Variation (CV)**

The CV is the ratio of the SEM to the mean; it expresses the spread of values around the mean as a percentage of it (e.g. the CV of a fatigue protocol is 10% of the total work performed).

**Limits of Agreement (LOA)**

The 95% LoA is a range where we can expect someone’s test-retest values to fall 95% of the time [5]. It is based on testing and can be calculated as follows:

Example:

The 95 % LoA for specific running task = ±7%

1^{st} test score = 10 min (600s)

Because of this, and providing no change has occurred, we can expect the retest time to be in the range of plus or minus 7% of the 1^{st} test score (10 min or 600s).

[Mean -7 % (Mean) to Mean +7 % (Mean)].

[600 – (7*(600/100) = 558s to (7*(600/100)) = 642s)]

As a result, we can expect (95% of the time) that the retest time will be between 9 minutes 18 seconds and 10 minutes 42 seconds. Practically speaking, this means that no change can be found to have taken place if retest values are within the above-mentioned range.

Understanding and testing reliability is relevant for both the practitioner and the researcher when selecting a measure [3], since it provides insights into the biological (e.g. circadian rhythm), environmental (e.g. wind speed), and/or technical (e.g. timing gate height) factors influencing score variance [7].

**For Practitioners/Coaches**

Upon starting with a new team, practitioners need to determine the purpose of the testing (e.g. classifying athletes as needing an intervention or tracking progress). After this, the focus should be to review the literature to find the best test for the envisioned purpose(s) (i.e. which test is most reliable and valid).

In this whole process, close attention should be given to specific factors of the study (e.g. sample characteristics: Regional vs. Olympic Athletes or testing procedures: timing gate height). In fact, if those factors are different in the practice than in the study, the reliability of the findings cannot be expected to be similar. For example, if a professional footballer was to perform a 40-m sprint test using timing gates it may render an ICC of 0.75, but the same 40-m sprint in high school athletes using a stopwatch will be much lower. This demonstrates the varying levels of reliability for the same test using different athletes and different equipment.

Finally, strict adherence to the procedures described in the supporting literature (e.g. equipment, test administrator, technical procedures, and familiarisation amongst many) is the key to best practice where results are reliable and valid [8].

**For Researchers**

In research, reliability is a useful tool to review the literature and help with study design.

Firstly, knowing about reliability will give insights into the relevance of results reported in the literature. For example, one can relate the change observed in an intervention study (e.g. +10%) to the reliability of the testing protocol used or cited. If the CV of the test is ± 6%, a retest value of +10% is clearly within the range where we can feel confident that no change has occurred. The fact that it reached statistical significance only demonstrates sufficient statistical power, not clinical significance.

Secondly, to estimate sample size, a well-designed study should account for the precision of the measurement used [5, 9-11]. The less precise the measurement, the larger the sample size will have to be in order to have enough statistical power to see a significant effect.

In order to have reliable results which can be used to implement coaching strategies or to publish as scientific literature, the following rules and procedures need to be implemented and documented:

**Train your testers**

- Document their training (e.g. duration and nature)
- Assess learning outcomes (e.g. checking mastery of testing procedure)

**Familiarise participants**

- Use reference protocols (e.g. standardised warm-up)
- Familiarise to your test (e.g. conduct a ‘dummy run’ of the test before the study officially starts)

**Implement the actual test as a pilot study**

- Analyse the results (i.e. Inter-tester reliability, Data reliability)

**Address potential issues **

- Modify testers training
- Address design issues (e.g. not enough rest time or unrealistic protocol)

In the previous section we defined reliability and highlighted its importance, in the current section we will present the statistical methods used to gauge it, but first let us briefly introduce some of the statistical assumptions that we will refer to later on.

**Assumptions**

Statistical testing is based on assumptions. In case of deviations, appropriate measures need to be taken else inappropriate results and conclusions might be drawn [12].

**Normality of distribution**

In a typical, normally distributed data set, a centred bell curve (Figure 2) demonstrates that 95% of the cases revolve around the mean by ±2 Standard Deviations [13]. In other words, when the data appears in a bell-shaped curve around the centre of the graph – as in Figure 2 – it suggests that 95% of the data revolves around the mean by ±2 Standard Deviations. In this case, the normality of distribution can be assumed.

Distribution normality can be assessed visually or by using significance tests in software packages [12]. However, the distribution is not considered ‘normal’ if the bell curve is too pointy, too flat, or skewed (Figure 3).

**Homoscedasticity**

The extent to which the size of the error (test-retest difference) is independent (i.e. not correlated) to the size of the tested value. Therefore, homoscedasitcity is when the test-retest difference is similar for people who score high and for people who score low.

For example, during ergometer testing [5]:

- Stronger athletes (SA) have higher power outputs (453.1 W)
- Weaker athletes (WA) have smaller power outputs (378.6 W)

**Homoscedasticity**: SA have similar test-retest differences than WA (6.2 W vs. 6.1 W) (left part of the Figure 4).

**Heteroscedascity**: SA have greater test-retest differences than WA (6.1 W vs. 4.4 W) (right part of Figure 4).

A change in the group mean across testing occasions (tested with a T-test or ANOVA) can signify a systematic error which, depending on its size, maybe a sign of design issues (e.g. an inconsistent testing environment). The acceptable size of the shift is decided by the researcher or the practitioner; however, it should really be as low as practically possible.

Although it is outside the scope of this article, it is interesting to note that the test of significance is influenced by the amount of random variation between tests, conversely, ANOVA and T-test’s assume, but are not very sensitive to, variations in normal distribution.

**Calculations**

The determination of the significance for the ANOVA is based on the F ratio, calculated as follows:

*F ratio = Between-day variance / Within-day variance*

The F ratio describes the separation between the scores across the days. In the diagram below we can see a high F ratio, due to a good separation between the days (6, 10, 14) and a rather low variation within each day. Statistical packages will give a statistical significance value for this test.

**Assumptions**

Normality of distribution and homoscedasticity are assumed but ANOVA is generally considered to be quite robust to violations if the sample is relatively large [14].

**Interpretation**

The practically acceptable bias size should be as low as possible, with the practitioner being the final judge of its appropriateness based on their expert opinion.

As previously mentioned, a change in the group mean across testing occasions suggests a systematic error which, depending on its size, maybe due to design issues:

**Decreasing group mean**: Due to fatigue carryover or lack of control of testing procedures. For example, inconsistency in the encouragement has a ±5% effect on strength performance [15].**Increasing group mean**: Due to a potentiating, training, or learning effect. For example, IQ scores increase with repeated tests [16].

**Limitations**

When sample size increases, the random error component will tend to cancel out across participants (it is random, so eventually it cancels out). In turn, there is an increased probability of statistical significance despite minimal differences between the days [5]. However, with ANOVA, the practitioner is free to act or not upon finding a statistically significant shift depending on his or her perception of the importance of the results.

Correlation measures assess the extent to which ‘Day 1’ and ‘Day 2’ measurements are associated. To do so, it compares the variation in data on the same day across the group to the sum of all variances.

**Calculations**

ICC = Between-day variance / (Within-day variance + Between-day variance)

The table below illustrates how the ICC changes when the source of variance changes. For example, true score variance decreases when ICC gets closer to 0.

When ICC is 0.8, 80% and 20% of the total variance are explained by true score and error variances, respectively.

**Assumptions**

Normality of distribution is assumed as in most parametric tests and similarly to the ANOVA since they have similar formulas.

**Interpretation**

The ICC is geared toward classification applications [3]. For example, classifying healthy versus pathologic shoulders when using a shoulder rotation test [2]. Several guidelines have been given for the classification of correlation coefficients [17].

**Poor correlation**: < 0.50**Moderate correlation**: 0.50-0.75**Good correlation**: 0.75-0.90**Excellent correlation**: > 0.90

**Limitations**

The biggest limitation of the ICC is that it does not completely describe the relationship between the two variables, this is because it does not account for the slope of the line formed by the test-retest points (as illustrated below in **Figure 6**). For example, all four data sets in Figure 6 have an ICC of 0.86.

Also, the ICC has a tendency for overestimation when calculated on *hetero*geneous samples [18], and is therefore not the best tool to compare separate studies [5]. Finally, it does not represent agreement between testing occasions [19].

While the previous sections discussed inter-individual comparisons (i.e. variation between participants), this section will focus on the variation associated with just one participant (i.e. the within-participant variation).

**Standard Error of Measurement (SEM)**

**Calculations**

The error of measurement is the root mean square average of each individual standard deviation as exemplified below in Figure 7.

The overall SEM is calculated as the root mean square average of all standard deviations:

SEM = √ ((5.7² + 4.2² + 5.7² + 2.8² + 2.1²) / 5)} = 4.3

**Assumptions**

No specific assumptions.

**Interpretation**

For a more practical outlook, practitioners can base their judgement of appropriateness on confidence intervals (CI) calculated as shown below:

- 68% CI = Score ± SEM
- 95% CI = Score ± (1.96 * SEM)
- 99% CI = Score ± (2.58 * SEM)

For example, if isokinetic quadriceps strength measured is 140 Nm and the specific test SEM is 15 Nm, statistical theory tells us that we can be 68% confident that true score is between: (140 – 15 = 125 and 140 + 15 = 155).

Based on an actual result of = 125 Nm < Quadriceps Strength < 155 Nm, the practitioner can assess whether the test is reliable and precise enough for the specific purpose and outcome.
**Limitations**

The SEM is expressed in the measured unit (e.g. distance or Watts), so that:

- The characteristics of the participants need to be accounted for when comparing the same test across different populations [5].

- The dimension of the test values (calibration or scaling across ergometers and analysers) needs to be accounted for when comparing the same population across different tests [5], as differences in scales will affect SEM size.

**Coefficient of Variation (CV)**

**Calculations**

The CV can be easily calculated using the following formula:

CV = SEM / mean * 100

Using the example from the previous section and the data from Figure 7, the CV can be calculated as follows:

(SEM = 4.3, Mean = (36 + 38 + 38 + 41 + 39.5) / 5 = 38.5)

CV = (4.3 / 38.5) * 100 = 11.17%

CV = 11.17%

**Assumptions**

CV assumes homoscedasticity after accounting for the mean, population of tests for each individual, as well as normality of distribution.

**Interpretation**

Hopkins [5] uses an ergometer based test to demonstrate the relevance of CV. Two groups of strong (SA) and weaker athletes (WA) perform the same test. The table below shows the results

We can see that the test appears to have lower reliability for the SA than for WA, until we examine the CV which actually renders the opposite picture and shows that the reliability of the test is similar for SA and LSA.

CV is a dimensionless statistic and has great appeal to compare the reliability of different measurement tools [5]. Some have suggested 10% or less as an acceptable value for CV [21], however, the decision and rationale belongs to the practitioner and will be largely a function of the testing aims.

With the CV, we can also use confidence intervals as described for the SEM: a CV of 10% means that [9] we can be 68% sure that a true test score will be between a measured value ±10% of the mean.

**Limitations**

The graph below shows that the lower the CV, the tighter the spread around the mean. Meaning that practitioners should be aware of the difference in precision that having an increase of 0.15 in CV induces.

**Limits of Agreement (LOA)**

Bland and Altman who introduced this measure, thought that looking at the range where an individual test score would fall 95% of the time may be more relevant than comparing test and retest.

**Calculations**

The Bland–Altman method is a two-step plotting process:

- Plot data points of each participant:
- x-axis: mean of the test-retest
- y-axis: difference of the test-retest

- Draw three lines representing group data:
- The Mean: difference between test and re-test (the ‘bias’)
- The Upper limit: The Mean + 1.96 * SD of test re-test differences
- The Lower limit: The Mean – 1.96 * SD of test re-test differences

**Assumptions**

This method assumes normality and homoscedasticity. It is recommended that a large sample size (*n* > 40) is used to avoid bias and facilitate comparisons between studies [5].

**Interpretation**

On the basis of two trials with X number of participants, when we test and re-test another participant the score in the second trial has a 5% chance, or 1 chance in 20, of being correct [5]:

> Score of 1^{st} test + the Upper Limit

OR

< Score of 1^{st} test – the Lower Limit

**Limitations**

LoA are considered by some as a reference when looking at absolute agreement between two testing occasions [9], yet it is largely disregarded by others [5]. Hopkins [5] argues that the 95% LOA maybe not be precise enough to monitor changes in performance in elite athletes. Hopkins [5] also argues that the 95% LoA are sensitive to sample size and show decreasing values with increasing sample size. Arguably, they might not be the best indicator of precision for one single trial as their main purpose is to provide a range in which the value of a re-test is expected to fall based on a test [5].

To conclude, there is not one best approach, and the practitioner/researcher should look carefully at both the absolute and relative reliability of a test and possibly present an extensive analysis in a detailed yet readable format to allow for comparisons. Care should be taken when extrapolating the findings of reliability studies, since findings are test- and sample-specific [23].

Field testing is the key to detect the worthwhile and externally valid effects that are the focus of applied research [24, 25]. However, precision of findings is altered due to the lack of control on extraneous variables and to the multiplication of error sources (e.g. low sampling rate) [24, 25]. To address this issue, we suggest studying the reliability and validity of applied research methods. Separating the different sources of error will help find practices that could increase precision (e.g. creating interpolation techniques to replace missing data).

We know that a familiarisation protocol decreases systematic bias in reliability studies, however, we do not know precisely why, or how, it works. Future investigations should examine the mechanisms which lead to test improvements observed following familiarisation for specific tests (e.g. decreased apprehension in a new environment like the laboratory or performing in front of a group). This will give better insights into the aspects that should be of focus, and will potentially create a roadmap to improve the effectiveness of familiarisation.

Reliability is the study of testing error and can be assessed using various statistical methods. Rather than common practices, research means and outcomes should guide the choice for the practitioner. Practically, the investigation of differences between groups or between time points must be preceded on a careful study of the reliability for the envisioned testing means.

Together with a very homogenous group, a reliable test will increase the chances of finding test-retest differences for a training intervention.

Some coaches believe that reading one article will make them an expert on Statistics. Here’s why they’re wrong…

Statistics entails many, many topics. By choosing to simply read up on Reliability and ignore the sea of other crucial topics surrounding statistics, you run the risk of being detrimental to your athlete’s success and not realising your full potential.

To make you an expert coach and make your life as easy as possible, we highly suggest you now check out this article on** Statistical Significance.**

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Jérôme Sauret

**Dr. Jérôme Sauret** PhD MSc BSc

Jérôme graduated in 2011 from the University of North Carolina at Greensboro (USA) with a PhD in Kinesiology and a minor in Statistics, after a BSc (Hons) and an MSc (Res) at the University of Gloucesterhire (UK). He is currently acting as a Movement and performance coach for successful entrepreneurs in the region of Montpellier as well as a lecturer in Statistics and performance in several MSc programs. Avid movement-based fitness practitioner and coach, his focus is to improve function by better understanding individual specificities in performance and training responses.

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