Civil Service Numerical Reasoning Test Statistics and Estimation in Context Syllabus

What is a frequency in the context of a data table?

The number of times a particular value or category occurs in the dataset.

What does $\sum f$ represent in a frequency table?

The total frequency, i.e. the total number of data items $n$ in the dataset.

In a frequency table, what does the $fx$ column give you and why is it useful?

It gives the value multiplied by its frequency, $f \times x$. Summing this column gives $\sum f x$, the total of all data values, used to find the mean.

What is cumulative frequency?

A running total of frequencies, adding each class frequency to the sum of all previous frequencies; useful for finding medians and quartiles from grouped data.

From a frequency table, how do you find the mean?

Compute $\sum f x$ and $\sum f$, then $$\bar{x} = \frac{\sum f x}{\sum f}$$

From a simple (ungrouped) frequency table, how do you locate the median value?

Find the median position $\frac{n+1}{2}$, build up cumulative frequencies, and identify which value's cumulative frequency first reaches that position.

When comparing two datasets, which two types of measure should you always compare?

A measure of average (location), such as the mean or median, and a measure of spread, such as the range or IQR.

If dataset A has a higher mean than dataset B, what does this tell you?

That values in A are typically larger/higher on average than those in B (A has a greater central value).

If dataset A has a smaller IQR than dataset B, what does this tell you?

That A is more consistent (less variable) — its middle 50% of values are clustered more tightly than B's.

When writing a comparison of two datasets, why must comparisons be made in context?

So conclusions relate to the real situation (e.g. higher scores = better performance), making the average and spread meaningful rather than just numerical.

For a fair comparison of two datasets of very different sizes, what should you compare instead of raw frequencies?

Relative measures such as proportions, percentages, or relative frequencies, since raw counts are not comparable across different totals.

What is relative frequency and how is it calculated?

The proportion of times a value occurs: $$\text{relative frequency} = \frac{\text{frequency of value}}{\text{total frequency}}$$

In a numerical reasoning test, why is estimation useful before doing a full calculation?

It quickly narrows the multiple-choice options to a plausible range, saving time and helping eliminate clearly wrong answers.

How can rounding values help you estimate an answer to shortlist options?

Round each value to convenient numbers (e.g. nearest 10 or 100), do the simpler arithmetic, and pick the option closest to the estimate.

To estimate $\frac{612 \times 19}{4}$, what rounding would give a quick approximation?

Round to $\frac{600 \times 20}{4} = \frac{12000}{4} = 3000$, so the answer is roughly $3000$.

What does it mean to 'sense-check' a statistical result?

To judge whether the answer is reasonable in context — checking magnitude, sign, and units — rather than accepting it blindly.

Why must a mean always lie between the smallest and largest data values?

Because the mean is a balance point of the values; a mean outside the data range indicates a calculation error.

If a question asks for an average test score (out of 100) and you calculate 340, what should you conclude?

The result is impossible (a score cannot exceed 100), so there is a calculation error that must be found and corrected.

When sense-checking a percentage, what range must the answer fall within (for a part of a whole)?

Between 0% and 100%; a value outside this range (for a simple proportion) signals an error.

What is the 'working backwards' strategy in a multiple-choice numerical test?

Substitute each given answer option into the problem to see which one satisfies the conditions, rather than solving the equation directly.

When is working backwards from answer options most efficient?

When direct algebra is slow or awkward but checking a candidate value is quick, and there are only a few options to test.

If you know the mean of 5 numbers is 12, how do you work backwards to find their total?

Multiply the mean by the count: total $= \bar{x} \times n = 12 \times 5 = 60$.

Four numbers have a mean of 9. A fifth number is added and the new mean is 10. What is the fifth number?

Old total $= 9 \times 4 = 36$; new total $= 10 \times 5 = 50$; the fifth number $= 50 - 36 = 14$.

How is the mean affected if every value in a dataset is increased by a constant $k$?

The mean also increases by $k$: $\bar{x}_{\text{new}} = \bar{x} + k$. The range and IQR are unchanged.

How is the mean affected if every value in a dataset is multiplied by a constant $k$?

The mean is multiplied by $k$: $\bar{x}_{\text{new}} = k\bar{x}$. The range and IQR are also multiplied by $k$.

What does it mean if the mean is much greater than the median in a dataset?

The distribution is positively skewed (skewed right) — a few large values pull the mean above the median.

In a comparison, which measure of average is fairer when the two datasets have very different shapes or outliers?

The median, because it resists distortion from outliers and skew, allowing a fairer like-for-like comparison of typical values.

A dataset is $4, 7, 7, 9, 13$. State its mean, median, mode and range.

Mean $= \frac{40}{5} = 8$; median $= 7$ (middle value); mode $= 7$; range $= 13 - 4 = 9$.

Why might the modal class not contain the median in a grouped distribution?

The modal class is where frequency is highest, while the median is the position-based middle of the data; these can fall in different classes depending on how data accumulates.

When estimating to eliminate options, what should you check about the order of magnitude of your estimate?

That it has the right number of digits/size; if options differ by powers of 10, a rough estimate of magnitude alone can eliminate most wrong answers.