Question

Scientists recorded the dry weight (in \(\mathrm{mg}\) ) of a species of flower growing in two habitats: unfragmented (natural) and fragmented [on land disturbed by human development (e.g., roadside or near a parking lot)]. The table below shows summary values. \({ }^{53}\) $$ \begin{array}{|lcc|} \hline & \text { Fragmented } & \text { Unfragmented } \\ \hline \text { Mean } & 86.2 & 92.6 \\ \text { SD } & 17.2 & 18.3 \\ n & 86 & 116 \\ \hline \end{array} $$ Calculate the sample effect size from these data.

Short answer

To calculate the sample effect size, first compute the pooled standard deviation, then use it to find Cohen's d value from the given means.

Step-by-step solution

Understand the Concept of Effect Size

The effect size is a quantitative measure of the strength of a phenomenon. In this case, the effect size will show the magnitude of the difference between the two habitat conditions for the flower's dry weight.

Calculate the Pooled Standard Deviation

To calculate the effect size, we first need the pooled standard deviation. The pooled standard deviation (Sp) is calculated using the standard deviations (SDs) and sample sizes (n) of both groups.\[Sp = \sqrt{\frac{(n_1 - 1)\times SD_1^2 + (n_2 - 1)\times SD_2^2}{n_1 + n_2 - 2}}\]

Calculate the Effect Size

Once the pooled standard deviation is calculated, we can find the effect size (Cohen's d) using the means (\bar{x_1} and \bar{x_2}) of both groups along with the pooled standard deviation (Sp).\[Cohen's d = \frac{\bar{x_1} - \bar{x_2}}{Sp}\]

Plug in the Given Values

Substitute the given mean, standard deviation, and sample size values into the formulas to calculate the pooled standard deviation and effect size:\[Sp = \sqrt{\frac{(86 - 1)\times 17.2^2 + (116 - 1)\times 18.3^2}{86 + 116 - 2}}\]\[Cohen's d = \frac{86.2 - 92.6}{Sp}\]

Key concepts

Pooled Standard Deviation

Pooled standard deviation is a method to combine the standard deviations of two or more different samples when the samples suggest similar but not identical variances on a common scale. It is a vital concept in statistics, particularly when you are dealing with multiple groups that need to be compared.

For those tackling problems in statistics, understanding pooled standard deviation can be quite crucial. Here's why: when comparing groups, such as the weights of flowers from different habitats as in our exercise, it is essential to establish a common ground for variability. The formula for pooled standard deviation is:
\[Sp = \sqrt{\frac{(n_1 - 1)\times SD_1^2 + (n_2 - 1)\times SD_2^2}{n_1 + n_2 - 2}}\]
Where:

• \(n_1\) and \(n_2\) are the sample sizes for the first and second groups, respectively.
• \(SD_1\) and \(SD_2\) are the standard deviations for the respective groups.

This statistic doesn't just jumble together the variances; instead, it accounts for the size of each group, giving you a balanced measure of spread.

Cohen's d

Cohen's d is a statistic used to measure the effect size, or in other words, the magnitude of the difference between two groups. It provides a standardized difference between means, which is especially helpful for comparing results across different studies or experiments.

To get to grips with Cohen's d, here are a few key points:

• It's a unitless measure, meaning that it allows for comparison across different units of measure.
• Interpreting Cohen's d depends on its value:
  • A small effect is typically around 0.2.
  • A medium effect is around 0.5.
  • A large effect is 0.8 or above.

The formula to calculate Cohen's d is quite straightforward:\[Cohen's d = \frac{\bar{x_1} - \bar{x_2}}{Sp}\]
With \(\bar{x_1}\) and \(\bar{x_2}\) representing the means of group one and two respectively and Sp as the pooled standard deviation. This way, Cohen's d normalizes the difference between means, giving us insights into the actual size of the effect beyond simple statistical significance.

Quantitative Measure

In the context of effect size, a quantitative measure can be thought of as a numerical characterization of the magnitude of a phenomenon. Unlike qualitative measures, which describe categorical data, quantitative measures deal with numbers and provide concrete evidence to support conclusions.

Moreover, quantitative measures like the effect size offer several advantages in research and analysis:

• They allow for clear and objective comparisons between groups.
• They facilitate the synthesis of results from different studies in meta-analyses.
• They provide a means to assess practical, as well as statistical significance.

Effect sizes are quantitative measures that capture the intensity of an effect without being swayed by sample size, which can be particularly important in fields where the scales of measurement can differ significantly. This objective approach allows for a more nuanced understanding of the data and emphasizes the importance of measuring the effect itself, rather than just looking at whether an effect exists.

Sample Size

The sample size, denoted by 'n' in statistics, refers to the number of individual observations or units included in a sample. It plays a crucial role in determining the reliability and precision of statistical measures, including effect size.

Here's why the sample size matters in statistical analysis:

• Statistical Power: A larger sample size increases the statistical power of a test, which means a better chance of detecting an effect if one exists.
• Confidence Level: Larger samples reduce the margin of error, offering tighter confidence intervals, thus enhancing the credibility of the findings.
• Variance Estimation: A greater sample size allows for a more accurate estimation of the population variance.

In practice, when calculating pooled standard deviation or Cohen's d, the sample sizes of the groups under comparison are fundamental to obtaining accurate measurements. As seen in the original exercise, the different sample sizes of the two habitats for the flower's dry weight are directly incorporated into the calculations, ensuring a fair representation of each group's impact on the overall variance and effect size.