Source: ENGR 216 Comprehensive Exam Practice Bank
Tags: central tendency, mean, median, mode, histogram, frequency distribution, bins, classes, probability, experimental data, peanut diameter, ENGR 216, Texas A&M
Central tendency describes the "middle" of a dataset using the mean, median, or mode. Histograms and frequency distributions organise raw experimental data into bins, letting you visualise spread and calculate the probability of sampling a value from a given range. These are foundational tools for every later topic in the course.
Central tendency
A single value that summarises where the centre of a dataset lies. The three standard measures are mean, median, and mode.
Mean (arithmetic mean)
The sum of all data values divided by the number of values. Often called the "average" in everyday language.
Median
The middle value when data are sorted in ascending order. For an even number of values, it is the average of the two middle values.
Mode
The value (or values) that appear most frequently in a dataset. A dataset can be unimodal, bimodal, or multimodal.
Theoretical centre
The term used in the context of frequency distributions and histograms to describe the "average" or the value representing the centre of a data set distribution. This is the specific terminology ENGR 216 uses on exams.
Frequency distribution
A table (or function) showing how often each value or range of values occurs in a dataset.
Histogram
A bar chart representing a frequency distribution. Each bar spans one class (bin), and its height corresponds to the frequency or relative frequency of that bin.
Class / bin
An interval that groups data values together in a frequency distribution. Accepted practice is to use no fewer than 6 and no more than 15 bins.
Relative frequency (probability)
The proportion of the total dataset that falls within a given bin. Calculated as the count in the bin divided by the total number of observations.
Mean, mode, and median are all valid descriptors of central tendency.
In experimental physics contexts, the mean is used most often, but the median is more robust to outliers, and the mode reveals the most common observation.
When a frequency distribution or histogram is involved, the term "theoretical centre" is used to refer to the average value representing the centre of the distribution.
When building a frequency distribution from experimental data, use a minimum of 6 bins and a maximum of 15 bins. This range keeps the distribution informative without being too coarse or too granular.
Each bin should have equal width.
The total area under a properly normalised histogram equals 1 (i.e. 100% of the data).
The probability of sampling a single item from a specific bin is:
$$P = \frac{\text{count in bin}}{N}$$
where N is the total number of observations.
Worked example (peanut diameter):
Given N = 231 peanuts and 33 peanuts in the 20–22 mm bin:
$$P = \frac{33}{231} = 0.142857... \approx 0.14$$
If you know the probability of a characteristic and receive a new sample of size n, the expected count is:
$$\text{Expected count} = P \times n$$
Worked example:
Probability of a peanut having major diameter 16.00–19.99 mm = 68.40% = 0.684. New sample size = 2000.
$$\text{Expected count} = 0.684 \times 2000 = 1368$$
Formula | Use |
|---|---|
$\bar{x} = \frac{1}{N}\sum x_i$ | Arithmetic mean |
$P(\text{bin}) = \frac{\text{count in bin}}{N}$ | Probability from frequency distribution |
$\text{Expected count} = P \times n$ | Predicting count in a new sample |
Bin count rule: minimum 6, maximum 15 classes.
⚠️ The exam uses the phrase "theoretical centre" to mean the average of a distribution. Know this specific term.
⚠️ When asked for probability from a histogram, always divide count by total N, not by the number of bins.
⚠️ Bin rules (6–15) are tested as a standalone recall question.
⚠️ Expected count questions give probability as a percentage. Convert to a decimal before multiplying.
Q: True or False: The mean, mode, and median are all valid descriptors of central tendency in experimental data analysis.
A: True. All three describe the centre of a dataset in different ways.
Q: What specific term describes the "average" or centre of a data set distribution in the context of frequency distributions and histograms?
A: Theoretical centre.
Q: What is the accepted rule for the minimum and maximum number of bins in a frequency distribution?
A: No fewer than 6 and no more than 15.
Q: A dataset contains 231 peanuts. 33 fall in the 20–22 mm range. What is the probability of sampling one peanut from that range?
A: P = 33 / 231 = 0.14.
Q: The probability of a peanut diameter falling between 16.00 and 19.99 mm is 68.40%. In a batch of 2000 peanuts, how many would you expect in that range?
A: 0.684 × 2000 = 1368.
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