Reason About Data and Draw Conclusions From Them - MCAT Chemical and Physical Foundations of Biological Systems

A maximum effect is reached (system saturation). A plateau in a dose-response curve occurs when further increases in dose yield no additional response due to receptor or system saturation.

Logarithmic axis scaling. Logarithmic scaling compresses exponential relationships, making equal ratios appear as equal intervals on the axis.

$\frac{y_2 - y_1}{x_2 - x_1}$. The slope represents the rate of change in $y$ per unit change in $x$ between two points.

$\frac{x_1}{x_0}$. Fold change measures the ratio of a new value to the baseline, indicating multiplicative increase or decrease.

$\frac{x_1 - x_0}{x_0} \times 100%$. Percent change quantifies the relative difference from an initial value, expressed as a percentage.

No linear association is present between the variables. A correlation coefficient of $r=0$ implies no linear relationship exists between the variables.

As $x$ increases, $y$ tends to decrease (negative correlation). A strong negative linear trend in a scatterplot indicates that the variables move in opposite directions, supporting a negative correlation.

As $x$ increases, $y$ tends to increase (positive correlation). A strong positive linear trend in a scatterplot indicates that the variables move in the same direction, supporting a positive correlation.

Association only; causation cannot be inferred from correlation alone. Correlation establishes association but requires additional evidence, such as controlled experiments, to infer causation.

The group means likely differ significantly (often $p < 0.05$). Non-overlapping 95% confidence intervals suggest the true means are different, typically indicating statistical significance at p<0.05.

Probability of results at least as extreme as observed, assuming $H_0$. The p-value quantifies the evidence against the null hypothesis by calculating the probability of observing the data or more extreme under H0.

It reflects data spread, not uncertainty in the estimated mean. Standard deviation error bars show the variability within the sample data, whereas SEM indicates the uncertainty of the mean estimate.

The trend is tentative; low $n$ limits reliability and generalizability. Small sample sizes reduce statistical power, making observed trends less reliable and harder to generalize to the population.

$2$. The slope is calculated as the change in y divided by the change in x between the two points.

$25%$. Percent increase is derived from the relative difference between the new and old values, multiplied by 100%.

$4$-fold. Fold change is the ratio of the final value to the initial value, indicating how many times the signal has multiplied.

$y \propto x^{-2}$. A linear log-log plot with slope -2 indicates an inverse square relationship between the original variables.

Not statistically significant at $\alpha = 0.05$; fail to reject $H_0$. A p-value greater than the significance level α=0.05 provides insufficient evidence to reject the null hypothesis.

Median. The median, as the middle value, remains stable despite extreme values, unlike the mean which shifts toward outliers.

Greater variability; data are more dispersed around the mean. Standard deviation measures the average distance of data points from the mean, with larger values indicating wider dispersion.

$\frac{s}{\sqrt{n}}$. SEM estimates the precision of the sample mean as an approximation of the population mean, decreasing with larger sample size.

$95%$ of such intervals would contain the true mean over repetitions. In frequentist statistics, a 95% confidence interval means that if the experiment is repeated many times, 95% of intervals will capture the true population mean.