Confidence Interval: The Range Your Estimate Allows

A confidence interval takes four numbers, a mean, a spread, a count and a confidence level, and returns a range for the true mean. The build divides the spread by the square root of the count to get a standard error, multiplies the standard error by the level's multiplier, and lays the result either side of the mean. Run on the made up fifty month record, it returns minus 0.88 to 1.88 per cent.

Work out an interval on any four numbers

A mean, a spread and a count go in, a level is chosen, and every rung of the build is printed separately, so every rung can be checked on paper. The defaults hold the made up fifty month record described under the next heading, and on those defaults the interval runs from minus 0.8788 to 1.8788 per cent.

Somebody hands over a stack of monthly figures and asks for the average. Added and divided, they give 0.50 per cent. A single number carries no width at all, so 0.50 per cent is the honest answer to the question asked and also the most confident sentence anybody will utter all week. The figure looks exactly as certain on fifty observations as it does on five. A confidence interval puts the width back. The same record, asked for two more facts about itself, hands back a pair of endpoints instead of a solitary figure. The whole build is four arithmetic operations long and every one of the four can be watched from start to finish, so a calculator is the right shape for it.

One invented setup carries the whole of what follows, and it deserves laying out completely before any reading gets produced. Call the object a traded unitAn invented label for something that has a quoted price and no promise that the price stays put. The four operations below use none of it, so who might issue such a thing, and on what terms, is left blank. and call this particular one the Nakshatra unit, a name that was made up for these notes and belongs nowhere beyond them. Each month it posts one monthly changeHow far a price finished a month from where it opened that month, written as a percentage of the opening figure. One reading per month, and no other kind is needed here., and that reading is only ever one of five numbers: minus 9.00, minus 4.00, 1.00, 6.00 and 11.00 per cent, weighted 0.08, 0.18, 0.48, 0.18 and 0.08 in that order. The five values and their five weights together are the entire generatorThe paper rule behind the readings. The rule fixes what can appear and how often, and the rule was written before a single month existed, so its properties are facts rather than estimates. Assembled in notes that come earlier.. Fifty months were then drawn out of it and stored as a tallyFive counts standing in for fifty readings, one count against each value that can appear. The order the months arrived in plays no part in any sum done here, so nothing is lost in the compression. of 5, 9, 25, 8 and 3. The tally of 5, 9, 25, 8 and 3 is the fifty month record, and the record is the only one computed on here.

The truth about this unit is available, which almost no real record ever is. The generator was typed out first, and only afterwards were fifty months drawn from it. Weighting each of the five values by the number sitting beside it and totalling the five gives the Nakshatra unit a true average month of exactly 1.00 per cent. The same operation on the squared distances gives a true spread of 5.00 per cent exactly. Neither figure was measured and neither is an estimate of anything. Both fall straight out of a definition. When the calculator returns a pair of endpoints, the truth can be set beside the answer and its position read off. A real dataset never permits that move.

Where the standard error comes from, and why adding observations tightens it at the rate they do, is covered separately under the sampling distribution. Testing a stated position against this record is covered separately under hypothesis testing. The full claim an interval makes, as against what these four operations entitle anybody to say out loud, is argued out under populations and samples.

What does this calculator actually compute?

Four numbers go in and one range comes out. The range is the set of values for the true average that the record entered is consistent with, at the level requested. Building that range takes four arithmetic operations, and the operations carry an assumption that the printed answer never shows.

One structural fact explains half of what the answer looks like, so fix it in mind immediately. The tool works out one number, a reach, and then uses that same number twice: once subtracted from the mean and once added to it. The entered mean therefore sits at the exact centre of the answer, always, and that is a fact about the arithmetic rather than a fact about the world.

The centring leaves something out. Nothing in the arithmetic puts the truth at the centre, and nothing in it makes the middle of the range likelier than the edges. Consider a shop assistant asked how long a delivery will take. Four days is a point. Three to five days is a range, and four sits in the middle of it for no better reason than that the assistant reached one day out on either side. The middle of a stated range shows where somebody started from, not where the parcel is.

Where does each of the four numbers come from?

One question is answered about each of the four inputs: where the number is found. The meaning of each input is settled elsewhere.

All four are supplied by the reader, so the calculator estimates nothing whatsoever. The calculator has never seen the record. Nor can it check that the entered mean came from the observations it is said to have come from, whether the count is honest, or whether 95 per cent was a sensible level to request. Divide, multiply, subtract, add: those four operations are the complete list of what it does.

One of those four rows is the row where two different answers are both available from the same record, so that row deserves a note. The spread should come from the corrected denominatorDividing the total of squared gaps by one less than the count rather than by the count itself. Which of the two to use, and why they differ, is settled in the notes on dispersion and is only named here.. The fifty month record gives 4.9744 per cent that way and 4.9244 per cent the other way, so the two do not agree, and feeding the wrong one in produces an interval that is quietly a shade too narrow. Which denominator belongs where is covered separately under dispersion. Knowing which of the two is in hand is the whole of what matters here.

How is the interval built, one visible step at a time?

A tool that prints only its answer cannot be checked, and a tool that cannot be checked has to be believed instead. So this one shows the standard error, then the multiplier, then their product, then both endpoints, and each of the five lines can be reproduced on paper in under a minute.

On the defaults, the spread of 4.9744 per cent is divided by the square root of the count, and the square root of 50 is 7.0711, so the standard error is 0.7035 per cent. The level of 95 per cent asks for a multiplier of 1.959964. Reaching 1.959964 standard errors out from the middle of the normal shapeThe familiar symmetric bell, assembled in notes that come before this one. The bell appears here only as the place the multiplier is read from, and nothing about it is put together again. leaves 2.5 per cent of the shape outside on each side. Multiply 0.7035 by 1.959964 and the reach is 1.3788 per cent. Subtract the reach from 0.50 and add the reach to 0.50, and the two endpoints are minus 0.8788 and 1.8788 per cent. The printed pair reads minus 0.88 and 1.88. A reader with a pen can reproduce every line above. Nothing else entitles a tool printing an answer to be trusted at all.

How wrong was the estimate, and where did the truth land?

A known generator pays for the comparison that follows. The mean this record produced is 0.50 per cent. The Nakshatra unit's own true average month, taken straight off the rule, is 1.00 per cent. So the point estimateOne number offered as the whole answer, carrying no width of any kind. The claims such a number does and does not make are argued out under populations and samples. was out by 0.50 percentage points, which is to say it was exactly half the truth. Read on its own, 0.50 per cent gives a badly wrong idea of what this unit does in an average month.

Now look at the two endpoints. The interval runs from minus 0.8788 to 1.8788 per cent, and 1.00 sits comfortably between them. The single number missed by half and the range around it still covered the answer, and that pairing is the entire reason this arithmetic exists. The record did not become better when the interval was drawn around it. The change is that the width stopped hiding.

One caution about how to read that, and it matters. Saying the truth landed inside is a description of what happened on this one record. The description says nothing about how often such a thing happens, or about the chance that the truth is inside any particular pair of endpoints. The argument about that chance is genuinely slippery, and it is settled separately, under populations and samples. The fact is checkable and the generator is on the table, so the fact is simply recorded here.

What happens to the interval as the count rises?

One relationship inside the tool is worth watching closely. The count is the only input where a reader's decision changes the answer in a way the formula does not make obvious. Hold the mean at 0.50 per cent, hold the spread at 4.9744 per cent, hold the level at 95 per cent, and change nothing but the count.

Reading the last column downwards: each row holds four times the observations of the row above it, and each width is half the width above it. Four times the record buys half the width, and it buys nothing at all in position. That exchange rate is worth carrying around in words rather than leaving it buried in the square root, because it is the answer to a question people ask constantly: how much more data would I need to halve this. The answer is four times as much, and then four times that again for the next halving.

The second half of that bold sentence is the half people forget. 0.50 per cent is the mean that was typed in, and no amount of counting can move it, so all four rows are centred there and the interval tightens around wherever the starting point was. A householder weighs the same sack of rice on the same shaky kitchen scale twenty times instead of twice, and the average of twenty readings wobbles far less than the average of two. The width is shrinking. But if the scale itself sits two hundred grams heavy, twenty readings and two thousand readings both centre on a figure two hundred grams heavy. A settled look is what makes the tighter answer the more dangerous of the two.

What is the answer assuming while it prints two decimal places?

Every reading this tool returns rests on one assumption, and the assumption is not in the record. The assumption is in the multiplier. A level of 95 per cent yields 1.959964 because that is how far out the reach must go before only 2.5 per cent of the normal shape is left on each side, and using it is a bet that the normal shape describes how a mean built from the record wobbles about the truth.

On fifty observations that is a reasonable bet, on eight it is a poor one, and the tool will return an answer at eight without a murmur of complaint. This is the part worth staring at. Nothing about the arithmetic changes between the two. Divide the spread by the square root of the count, reach out 1.959964 of those, subtract once and add once. The same four operations, in the same order, producing the same tidy two decimal places. The multiplier's right to be that number at all is what changed silently. The tool never saw the record and cannot count what it was not given, so it has no way to notice.

The consequence is specific: an interval that comes out too narrow, and reads as though it had been measured. The same mean and the same spread entered into the instrument with a count of eight return minus 2.95 to 3.95 per cent, printed to the same two decimals as everything else above. A range invites a firmer conclusion than the record can carry, and invites it in the language of precision, so a narrow interval on eight observations is a worse outcome than no interval at all. The repair is a habit rather than a formula: quote the count in the same sentence as the interval, every time. Written as minus 2.95 to 3.95 per cent the reading looks settled; written as minus 2.95 to 3.95 per cent on eight observations it tells its own story, and the next person to read it can decide what two decimal places deserve.

The second half of the caution is about the question rather than the arithmetic. Every operation above is about an average. The interval says nothing about where a single future month will land. A range for one future month is a different and considerably wider question, and it is covered separately. The record itself makes the point obvious. Within these fifty months, five landed on minus 9.00 per cent and three on 11.00, and neither number is remotely close to the range minus 0.88 to 1.88.

How is an interval handed over by somebody else checked?

Here is what a lender, an analyst or anybody signing off on arithmetic they did not perform actually does, and it takes about two minutes. A credit officer looking at a small business borrower whose monthly takings have been summarised into a range, or an analyst handed a supplier's average defect rate with a range attached, runs the same five checks. None of them require the underlying observations. All five can be done on the back of the sheet itself.

First, recompute the standard error: the quoted spread over the square root of the quoted count, and see whether it lands where they landed. Second, confirm the mean sits at the centre. The two endpoints added and halved must give the mean they typed in. Minus 0.8788 plus 1.8788 is 1.0000, halved is 0.5000, the record's mean exactly. Third, quadruple the count mentally and check that the width would halve. Fourth, ask which denominator the spread came from. The same fifty months yield 4.9744 or 4.9244 depending on the answer. Fifth, and most important, ask whether the question was ever about an average at all.

The fifth check saves the most work. If the question was about a single future case, then these four operations are simply the wrong tool, and no amount of correct arithmetic will fix that. Somebody asking what the average monthly change has been is asking a question this calculator answers. Somebody asking what next month will do is asking a different one, and it is covered separately. The two questions produce different ranges of very different sizes, and reaching for the narrow one because it was to hand is the most common way this arithmetic gets misused.

There is one more thing worth asking about, and it sits underneath all five checks. An estimatorThe recipe that turns a record into an answer, held apart from the answer any one record happened to produce. The word appears here just so the recipe and its output do not get muddled. is a recipe, and the interval is the recipe's output on one record. Two people running the same recipe on two different records will get two different intervals, and neither of them is doing anything wrong. The disagreement is the width being honest about itself.