Sport Science D.1

What is standard deviation (SD)?

It measures how spread out data points are around the mean, indicating whether a value represents a meaningful difference or normal variation.

What percentages of data lie within 1 and 2 SDs in a normal distribution?

68% of data lies within 1 standard deviation, and 95% lies within 2 standard deviations in a normal distribution.

How does SD help compare two or more groups?

It reveals consistency and variation between groups, where a smaller value indicates more consistent data and a larger value indicates more variation.

What are error bars?

They are a visual representation of variability or uncertainty, with longer bars indicating more variation and shorter bars more consistency.

Graph interpretation: Two athletes same mean, different SDs

The athlete with a smaller one exhibits more consistent performance.

Can SD be negative?

No, because it is based on squared deviations, which always result in a non-negative value.

How can SD and CV together give more insight?

Their combined use provides insight by showing both the absolute spread and the relative variability, which is better for comparing different datasets.

What is CV?

It is calculated as \text{(SD \div mean) \times 100\%} and demonstrates relative variability, where a higher value signifies more variation and a lower value indicates more consistency.

CV vs SD $-$ when is CV more useful?

It is more useful for comparing datasets that have different units of measurement.

Tricky exam example: Athlete A SD = 0.2, mean = 2; Athlete B SD = 0.4, mean = 5

Athlete B is more consistent relative to their mean, as their performance shows 8% variability compared to Athlete A's 10%.

What is IQR?

It is the range between the first and third quartiles (Q3 - Q1), representing the middle 50% of the data and focusing on central data by ignoring outliers, which is useful for box plots.

Why use IQR instead of SD?

Its use is preferred because it offers a more reliable measure of spread when outliers might distort standard deviation.

What is SE?

It quantifies how accurately the sample mean estimates the population mean, calculated as \text{SD \div \sqrt{n}}, where a larger sample size (n) leads to a smaller value and a more reliable mean.

SD vs SE?

One shows the spread of data points, while the other indicates the reliability of the sample mean.

Worked example: BlazePods reaction time

Given trials of 0.32, 0.34, 0.33, 0.35, 0.31 s with an uncertainty of \pm0.01 s, the average is 0.33 s, and the standard error (\text{SD \div \sqrt{n}}) is approximately 0.004 s, leading to a final result of 0.33 \pm 0.004 s.

How does increasing sample size affect SE?

A larger sample size (n) results in a smaller value, making the mean estimate more reliable.

Can SE be larger than SD?

No, because it is calculated as \text{SD \div \sqrt{n}} and will therefore always be less than or equal to the standard deviation.

Causes of uncertainty?

It can arise from random variation, measurement errors, and sampling bias.

How is uncertainty measured?

It is typically quantified using standard deviation, standard error, confidence intervals, and error bars.

Why is understanding uncertainty important?

Understanding it is crucial because it indicates reliability, helps identify errors and limits, and guides the formation of valid conclusions.

How do repeated trials reduce uncertainty?

Averaging the results from repeated trials reduces random error, which in turn decreases the standard error.

Connection: Error bars, SD, SE

Standard deviation describes the spread of data points, standard error demonstrates the reliability of the sample mean, and error bars provide a visual representation of overall variability.

Purpose of t-test?

Its purpose is to determine if observed differences between group means are statistically significant, meaning they are due to treatment effects rather than random chance.

Paired t-test?

It is used to compare connected data, such as measurements taken from the same individuals before and after an intervention.

Unpaired t-test?

It is used for comparing independent groups, like different sets of people in control versus experimental conditions.

Two-tailed t-test?

It investigates whether there is a difference between means without making an assumption about the direction of that difference.

How to determine significance in a t-test?

Significance is determined by comparing the calculated t-value to a critical t-value at a predetermined significance level, such as \alpha = 0.05.

Can a t-test show significance with similar means?

Yes, it can show significance even with similar means if the standard deviation is very low, indicating highly consistent data.

Worked example for paired t-test

An example involves comparing heart rate measurements taken from the same individuals both before and after exercise.

Worked example for unpaired t-test

An example involves comparing a training group to a control group without training to evaluate the effect of the intervention.

What is correlation?

It quantifies how two variables move together, indicating the strength and direction of their linear relationship.

What is causation?

It describes a relationship where one variable directly brings about a change in another variable.

Why does correlation not imply causation?

It does not imply causation because confounding factors or a third hidden variable might be influencing both variables, or the relationship could simply be coincidental.

What is the Coefficient of Determination (R^2)?

It reveals the predictive power of one variable over another by indicating the proportion of variance in the dependent variable explained by the independent variable, expressed as a value between 0 and 1 or 0% to 100%.

Distinguish between R^2, correlation, and a t-test.

Correlation measures the strength and direction of a linear relationship between two variables, the coefficient of determination (R^2) indicates one variable's predictive ability for another, and a t-test assesses the statistical significance of differences between group means.

Sport example: Football passing vs. goals scored

With an r = 0.75 and an R^2 = 0.56, this signifies a strong positive correlation, where 56% of the variance in goals scored can be predicted by passing statistics, though it does not imply causation.

Convert these numbers to scientific notation:

0.000006 converts to 6 \times 10^{-6}, 5,400,000 to 5.4 \times 10^{6}, 0.009 to 9 \times 10^{-3}, and 60 to 6 \times 10^{1}.

Multiply using scientific notation: (3 \times 10^{5}) \times (4 \times 10^{-2})

Multiplying (3 \times 10^{5}) by (4 \times 10^{-2}) yields 1.2 \times 10^{4} by first multiplying the coefficients and then adding the exponents.

Divide using scientific notation: (8 \times 10^{6}) \div (2 \times 10^{2})

Dividing (8 \times 10^{6}) by (2 \times 10^{2}) results in 4 \times 10^{4} by dividing the coefficients and subtracting the exponents.

Why is scientific notation used?

It is used to simplify the representation of extremely large or small numbers, making them more manageable for calculations and comparisons.

What is standardization in experimental design?

This concept involves establishing clear, repeatable experimental steps or protocols to ensure consistent and reliable results.

What is experimental control?

This process involves maintaining the stability and constancy of the experimental environment and non-studied variables to draw reliable conclusions about the variable being tested.

Why are standardization and control important in experiments?

They are important because they reduce unwanted variation, enhance data quality, and ensure fair comparisons between experimental conditions, thus leading to more valid and reliable conclusions.

How do standardization and control connect to t-tests?

Consistent experimental protocols, achieved through these practices, facilitate the acquisition of accurate standard deviations, which are essential for valid t-test results.

Can experiments without a control group still be valid?

While sometimes considered, experiments without one are generally less reliable and harder to interpret, as it becomes difficult to attribute observed changes solely to the intervention without a comparative baseline.

Athlete long jump results: 7.1 m, 7.3 m, 7.2 m, 7.4 m, 7.1 m.

Calculate the mean, SD, and CV, then interpret the consistency.

For results of 7.1 m, 7.3 m, 7.2 m, 7.4 m, 7.1 m, the mean is 7.22 m, the standard deviation is approximately 0.13 m, and the coefficient of variation is about 1.8%, which indicates very consistent performance.

In golf putting, Player A has an SD = 2 cm, and Player B has an SD = 5 cm. Who is more consistent?

Player A is more consistent because a smaller standard deviation indicates less variability in their putting.

Given 5 BlazePods reaction time trials with an average of 0.33 s and an uncertainty of \pm0.01 s. Calculate the average \pm SE.

Assuming the \pm0.01 s uncertainty represents the standard deviation, the average of 0.33 s across 5 trials plus or minus the standard error (\text{0.01 \div \sqrt{5}} \approx 0.004 s) yields a final result of 0.33 \pm 0.004 s.

In football, analyses show a correlation (r) between possession and goals scored of 0.75, with an R^2 of 0.56. Interpret these findings.

A correlation (r) of 0.75 indicates a strong positive linear relationship between possession and goals scored, and an R^2 of 0.56 means 56% of the variability in goals scored can be predicted by possession, although correlation does not imply causation.

Give two sport science examples each for when to use a paired t-test and an unpaired t-test.

Paired tests are used for comparing an athlete's sprint time before and after training or an individual's heart rate with varying equipment, whereas unpaired tests compare vertical jump height between different sport groups or evaluate strength gains in supplement versus control groups.

Given SD = 0.15 and mean = 5, calculate the Coefficient of Variation (CV) and interpret it.

With a standard deviation of 0.15 and a mean of 5, the coefficient of variation is calculated as 3%, indicating very low relative variability and consistent data points relative to the mean.

If a graph shows error bars for different athletes or groups, how would you identify which one is more consistent?

The athlete or group presenting shorter error bars is considered more consistent, as this visual cue indicates less variability or uncertainty in their data.