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Three columns compare the three document formats by what each one stores inside the file. An editable document such as a .docx stores structure: headings, paragraphs, lists and styles. Its text is real text, it is fully editable, and its page layout is computed when the file is opened, so a missing font is substituted and line and page breaks can move. A PDF stores a finished page: text placed at coordinates with fonts embedded. Its layout is fixed everywhere, editing is awkward by design, and its text is real text only when the PDF was exported from a document rather than made from a scan. An image such as PNG or JPEG stores a grid of pixels, with no words, headings or structure at all. It cannot be edited as a document, and it is pixel exact but never adapts, so enlarging it enlarges the blur rather than adding detail. A band below marks the single distinction that causes the most trouble: a PDF exported from a document and a PDF made from a scan or photo look identical on screen, yet only the first contains real text and the second is a picture in a PDF wrapper. The two second test on any file is to try selecting a sentence, because if nothing highlights then only OCR will put text into it. A closing note records that SVG is the exception among image formats, being a vector format whose text stays real text, and that accessibility is a property of the individual file rather than of the format, since a PDF is accessible only once it has been tagged.

PDF vs Word vs Image Files: Which Format Should You Use?

A practical decision guide to document formats: what each one actually stores, which ones keep text machine-readable, what that means for editing, accessibility, printing and archiving, and when converting is worth it.

Updated 15 Sept 202617 min readWeb Tools
A base value of 100 sits on a dashed horizontal line marking the 100 level, and forks into two paths. The upper path applies a 20 percent increase by multiplying by 1.20, adding 20 units taken from a base of 100 and rising to 120. The lower path applies a 20 percent decrease by multiplying by 0.80, removing 20 units also taken from a base of 100 and falling to 80. Each path then has the opposite change applied and both arrows converge on a single box reading 96, drawn below the dashed line so the shortfall is visible. From 120 the multiplier 0.80 removes 24 units rather than 20, because the percentage is now taken from 120, and from 80 the multiplier 1.20 adds only 16 units rather than 20, because the percentage is now taken from 80. A note beside the result explains that both routes land on 96, four short of the start: the first 20 percent was 20 units of 100, while the second was 20 percent of a different number, either 24 off 120 or 16 onto 80. A blue band gives the general result, that applying plus r then minus r leaves you at one plus r times one minus r, which equals one minus r squared, or 0.96 for a rate of 0.20, and that because r squared is always positive you always finish below the start, in either order and at every rate. An amber band separates percentage change from percentage points, noting that a move from 20 percent to 25 percent is a rise of 5 percentage points and also a 25 percent relative increase. A closing green band gives the reverse rule, original equals new divided by one plus the rate, warns that subtracting the percentage back does not work, and notes that percentage change from a base of zero is undefined because the formula divides by the original value.

Percentage Increase vs Percentage Decrease Explained

One formula, one base value, and the reasons percentage change trips people up: why +20% then -20% does not return you to the start, how to reverse a change, why percentage points are a different measurement, and why a base of zero has no answer.

Updated 15 Sept 202612 min readMath
The formula z equals x minus mu over sigma is set out at the top with its parts named: x is the value, mu is the mean of the reference group, sigma is the standard deviation, the result is a pure number with no units, and the sample form is x minus x bar over s. Below it a symmetric bell curve is drawn over a number line marked at minus 3, minus 2, minus 1, 0, plus 1, plus 2 and plus 3 standard deviations, with a dashed vertical line through the centre labelled mean, z equals 0. A second row under the axis translates those positions into the raw marks of a worked example where the class mean is 76 and the standard deviation is 8, so minus 1 is 68 marks, 0 is 76 marks, plus 1 is 84 marks and plus 2 is 92 marks. A marker on the curve picks out a score of 88 in that class at z equals plus 1.5. Text to the left records that a negative z is below the mean and is not a judgement, since for error rates or delays below average is the good direction, and text to the right records that a positive z is above the mean, that the sign gives the direction and the size gives the distance. A band beneath works the example through: 88 minus 76 is 12, divided by 8 gives plus 1.5, while the same score of 88 in a class whose mean is 84 and standard deviation is only 2 gives plus 2.0, a smaller gap but a bigger z. A blue band then states the distinction that matters most: calculating a z-score assumes nothing about the shape of the data, because it is arithmetic on a mean and a spread, whereas turning a z-score into a percentile or a probability assumes the standard normal distribution, and the familiar figures of 68, 95 and 99.7 percent belong to that curve rather than to z-scores. A closing amber band notes that thresholds such as absolute z above 2 or above 3 are rules of thumb rather than laws: they flag more points as a dataset grows simply because there are more points, far more values sit out in the tails of skewed or heavy tailed data than a normal curve predicts, and an extreme value inflates the very standard deviation used to judge it, so a real outlier can hide its own score.

Z-Scores Explained: What They Mean in Statistics

What a z-score is, how to calculate it from a mean and standard deviation, how to read positive and negative values, and the difference between standardising a value and claiming a percentile.

Updated 15 Sept 202614 min readMath

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