Question

Researchers measured the breadths, in \(\mathrm{mm}\), of the ankles of 460 youth (ages \(11-16\) ); the results are shown in the table. \(^{51}\) $$ \begin{array}{|ccc|} \hline & \text { Males } & \text { Females } \\ \hline n & 244 & 216 \\ \bar{y} & 55.3 & 53.3 \\ s & 6.1 & 5.4 \\ \hline \end{array} $$ Calculate the sample effect size from these data.

Short answer

Calculate the pooled standard deviation and use it in Cohen's d formula to determine the effect size from the given sample means and standard deviations.

Step-by-step solution

Understand and Identify the Data

First, identify the given data from the table: - The sample sizes are 244 males and 216 females. - The sample means are 55.3 mm for males and 53.3 mm for females.- The standard deviations are 6.1 mm for males and 5.4 mm for females.

Calculate the Effect Size Using Cohen's d

The effect size can be calculated using Cohen's d formula, which is: \[ d = \frac{\bar{x}_1 - \bar{x}_2}{s_{pooled}} \]where: \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, and \( s_{pooled} \) is the pooled standard deviation.

Calculate the Pooled Standard Deviation

The pooled standard deviation is calculated using the formula: \[ s_{pooled} = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \]Substitute the given values to find the pooled standard deviation.

Substitute Values into the Cohen's d Formula

Once the pooled standard deviation is found, substitute the mean values and the pooled standard deviation into the Cohen's d formula to find the effect size.

Interpret the Effect Size

Interpret the calculated effect size based on Cohen's guidelines. Generally, 0.2 indicates a small effect, 0.5 a medium effect, and 0.8 a large effect.

Key concepts

Understanding Effect Size and Cohen's d

Effect size is a statistical concept that measures the strength of the relationship between two variables. In the context of the exercise given, effect size tells us how significant the difference is between the ankle breadths of males and females. One widely used measure of effect size is Cohen's d, which provides a scale of standard deviations between the means of two groups.

The calculation of Cohen’s d is straightforward. By subtracting the average breadth of female ankles from that of male ankles and dividing by the pooled standard deviation, we obtain a dimensionless number. This number can be interpreted using Cohen's guidelines, which categorizes effect sizes as small (0.2), medium (0.5), and large (0.8). This interpretation can provide insights into the practical significance of the findings from the research study, beyond just statistical significance.

Pooled Standard Deviation Clarified

Pooled standard deviation is key to comparing groups with different sample sizes and variances, as seen in our exercise with different numbers of males and females. This composite measure gives us a way to estimate the standard deviation of combined groups.

To calculate this, we use the sample sizes and the standard deviations of both groups, obtaining an average standard deviation that is 'pooled' from both samples. The formula weights the variance (standard deviation squared) of each group by its degrees of freedom (number of observations minus one), which corrects for the fact that larger samples provide more reliable estimates. The square root of the weighted average gives us the pooled standard deviation, a critical component in calculating effect sizes like Cohen's d.

The Significance of Sample Means

Sample means represent the average values of our observations—in this case, the average ankle breadth for males and females respectively. These are crucial for understanding the central tendency of our data and for comparing the two groups. When calculating the effect size using Cohen's d, the difference in sample means reflects the magnitude of measurement differences between the two groups.
The calculation for the Cohen's d in our example relies on these averages, necessitating precise computation and understanding. The sample means work in tandem with the pooled standard deviation to provide context and scale to the effect size, informing us how much the groups differ on average in comparison to the variability within each group.

Standard Deviation Demystified

Standard deviation is a measure that indicates the average amount of variation or dispersion from the mean in a set of data. In the context of our ankle breadth study, it tells us how much the individual measurements within each group of males and females tend to deviate from their respective average values.

Different standard deviations between groups are common and expected, reflecting the individual differences within each gender. These values are squared and play a key role in computing the pooled standard deviation. Understanding both the absolute values of the standard deviations and the relative differences between them is essential for interpreting variability and consistency within our samples, and thus for understanding the full meaning of Cohen's d.