Examius
🇮🇳 GATE Ecology and Evolution · flashcards
GATE Ecology and Evolution Mathematics and Quantitative Ecology Flashcards
50 question-and-answer cards covering Mathematics and Quantitative Ecology as it is examined in GATE Ecology and Evolution. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics and Quantitative Ecology deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
State the multiplication rule for independent events.
If $A$ and $B$ are independent, then $$P(A \cap B) = P(A) \cdot P(B).$$
What does it mean for two events to be independent?
Two events are independent if the occurrence of one does not affect the probability of the other; formally $P(A \mid B) = P(A)$, equivalently $P(A \cap B) = P(A)P(B)$.
Define conditional probability $P(A \mid B)$.
$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$ the probability of $A$ given that $B$ has occurred.
What distinguishes mutually exclusive events from independent events?
Mutually exclusive events cannot occur together ($P(A \cap B) = 0$). Independent events can occur together, with $P(A \cap B) = P(A)P(B)$. Two events with nonzero probability cannot be both mutually exclusive and independent.
What is a frequency distribution?
A frequency distribution is a table or summary showing how often each value or class interval of a variable occurs in a dataset, displaying the counts (frequencies) across categories.
Define relative frequency.
Relative frequency is the proportion of observations in a class: $$\text{relative frequency} = \frac{\text{class frequency}}{\text{total number of observations}}$$
What are the three common measures of central tendency?
Mean (arithmetic average), median (middle value when ordered), and mode (most frequently occurring value).
State the formula for the arithmetic mean of a sample.
$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}$$
How are the median and mode defined?
The median is the middle value of an ordered dataset (average of the two middle values if $n$ is even). The mode is the value that occurs most frequently.
State the formula for the population variance $\sigma^{2}$.
$$\sigma^{2} = \frac{1}{N}\sum_{i=1}^{N} (x_{i} - \mu)^{2}$$ the mean of the squared deviations from the mean $\mu$.
State the formula for the sample variance $s^{2}$.
$$s^{2} = \frac{1}{n-1}\sum_{i=1}^{n} (x_{i} - \bar{x})^{2}$$ The divisor $n-1$ (Bessel's correction) gives an unbiased estimate.
What is the standard deviation and how does it relate to variance?
The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^{2}}$ (or $s = \sqrt{s^{2}}$). It measures spread in the same units as the data.
What is the coefficient of variation (CV) and its formula?
The coefficient of variation is a relative measure of dispersion: $$CV = \frac{\sigma}{\mu} \times 100\%$$ It expresses the standard deviation as a percentage of the mean, allowing comparison of variability between datasets with different units or means.
Why is the coefficient of variation useful compared to the standard deviation alone?
Because it is unitless (a ratio), the CV allows comparison of variability across datasets that have different units or very different means, which raw standard deviations cannot do fairly.
What does the correlation coefficient measure, and what is its range?
The (Pearson) correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It ranges $-1 \leq r \leq 1$.
Interpret the values $r = 1$, $r = -1$, and $r = 0$.
$r = 1$: perfect positive linear correlation; $r = -1$: perfect negative linear correlation; $r = 0$: no linear correlation between the variables.
State the formula for the Pearson correlation coefficient.
$$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^{2}\,\sum (y_i - \bar{y})^{2}}}$$
Why does correlation not imply causation?
A statistical correlation between two variables only indicates they vary together; it does not prove one causes the other. The association may be due to a confounding (lurking) variable, reverse causation, or coincidence.
What is a p-value?
A p-value is the probability of obtaining a test result at least as extreme as the observed one, assuming the null hypothesis $H_0$ is true. A small p-value indicates the data are unlikely under $H_0$.
How is a p-value used to make a decision at significance level $\alpha$?
If $p \leq \alpha$ (e.g. $\alpha = 0.05$), reject the null hypothesis $H_0$ (result is statistically significant). If $p > \alpha$, fail to reject $H_0$.
Define a Type I error and its probability.
A Type I error is rejecting the null hypothesis $H_0$ when it is actually true (a false positive). Its probability equals the significance level $\alpha$.
Define a Type II error and its probability.
A Type II error is failing to reject the null hypothesis $H_0$ when it is actually false (a false negative). Its probability is denoted $\beta$, and statistical power $= 1 - \beta$.
Compare Type I and Type II errors in a hypothesis test.
Type I error ($\alpha$): rejecting a true $H_0$ (false positive). Type II error ($\beta$): failing to reject a false $H_0$ (false negative). Decreasing $\alpha$ generally increases $\beta$, all else equal.
What is a test statistic in hypothesis testing?
A test statistic is a value computed from sample data (e.g. a $z$, $t$, $\chi^{2}$, or $F$ statistic) that measures how far the sample result deviates from what is expected under $H_0$. It is compared to a reference distribution to obtain the p-value or critical value for the decision.
What this deck covers
The Mathematics and Quantitative Ecology deck follows the GATE Ecology and Evolution Mathematics and Quantitative Ecology syllabus — 2 chapters and 9 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 25.0 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 162 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematics and Quantitative Ecology flashcards FAQ
How many Mathematics and Quantitative Ecology flashcards are in this GATE Ecology and Evolution deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these GATE Ecology and Evolution flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematics and Quantitative Ecology cards cover?
They follow the GATE Ecology and Evolution Mathematics and Quantitative Ecology syllabus — 2 chapters and 9 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.