[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"tool-content:en:standard-deviation-calculator":3,"glossary:en":56,"published-tools-en":105},{"id":4,"documentId":5,"slug":6,"intro":7,"howTo":8,"longContent":9,"createdAt":10,"updatedAt":11,"publishedAt":12,"locale":13,"name":14,"faq":15,"examples":40,"category":41,"seo":50,"localizations":55,"metaTitle":52,"metaDescription":53},370,"wfdwjimxrjcenp60lsgf8qq7","standard-deviation-calculator","\u003Cp>A \u003Cstrong>standard deviation calculator\u003C\u002Fstrong> measures how far a list of numbers sits from its own mean. The eight values loaded by default, 2, 4, 4, 4, 5, 5, 7, 9, return \u003Cstrong>2.13809\u003C\u002Fstrong> as a sample and \u003Cstrong>exactly 2\u003C\u002Fstrong> as a population. The page computes both divisors on every entry, next to the variance, the sum of squared deviations, the range and the deviation of each value.\u003C\u002Fp>","\u003Col>\u003Cli>Paste the numbers into the field, separated by commas, spaces, semicolons or line breaks. A column copied straight from a spreadsheet drops in unchanged, up to 500 values.\u003C\u002Fli>\u003Cli>Pick Sample (n − 1) or Population (n). Both results stay on screen either way, so the choice only decides which one the page shows large at the top.\u003C\u002Fli>\u003Cli>Read the standard deviation and its variance, then the count, the mean, Σ(x − x̄)², the minimum, the maximum and the range underneath.\u003C\u002Fli>\u003Cli>Open the deviation table to see which values carry the spread, then export the breakdown as CSV or TXT.\u003C\u002Fli>\u003C\u002Fol>","\u003Ch2 id=\"step-by-step\">How to calculate standard deviation step by step\u003C\u002Fh2>\n\u003Cp>Take the eight numbers the page loads by default, 2, 4, 4, 4, 5, 5, 7, 9: they add up to 40 across eight values, which puts the \u003Ca href=\"\u002Fmath\u002Faverage-calculator\">mean\u003C\u002Fa> at \u003Cstrong>5\u003C\u002Fstrong>, the pivot every later step measures against.\u003C\u002Fp>\n\u003Cp>Subtracting 5 from each number gives the deviations, running from −3 for the lowest value up to +4 for the 9. Those signed deviations always add back to zero, which is the reason each one gets squared first. The squares total \u003Cstrong>Σ(x − x̄)² = 32\u003C\u002Fstrong>. The single value 9 carries half of that total on its own.\u003C\u002Fp>\n\u003Cp>Dividing 32 by \u003Cstrong>n − 1 = 7\u003C\u002Fstrong> gives a sample variance of 4.571429, whose square root is the sample standard deviation \u003Cstrong>s ≈ 2.13809\u003C\u002Fstrong>. Dividing the same 32 by \u003Cstrong>n = 8\u003C\u002Fstrong> gives a population variance of 4 and a population standard deviation of \u003Cstrong>exactly 2\u003C\u002Fstrong>. The variance is that same figure one step earlier, still in squared units; the square root brings it back into the units of the data.\u003C\u002Fp>\n\u003Ctable>\u003Cthead>\u003Ctr>\u003Cth scope=\"col\">Value\u003C\u002Fth>\u003Cth scope=\"col\">x − x̄\u003C\u002Fth>\u003Cth scope=\"col\">(x − x̄)²\u003C\u002Fth>\u003Cth scope=\"col\">Share of the total\u003C\u002Fth>\u003C\u002Ftr>\u003C\u002Fthead>\u003Ctbody>\u003Ctr>\u003Cth scope=\"row\">2\u003C\u002Fth>\u003Ctd>−3\u003C\u002Ftd>\u003Ctd>9\u003C\u002Ftd>\u003Ctd>28.125 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">4\u003C\u002Fth>\u003Ctd>−1\u003C\u002Ftd>\u003Ctd>1\u003C\u002Ftd>\u003Ctd>3.125 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">4\u003C\u002Fth>\u003Ctd>−1\u003C\u002Ftd>\u003Ctd>1\u003C\u002Ftd>\u003Ctd>3.125 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">4\u003C\u002Fth>\u003Ctd>−1\u003C\u002Ftd>\u003Ctd>1\u003C\u002Ftd>\u003Ctd>3.125 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">5\u003C\u002Fth>\u003Ctd>0\u003C\u002Ftd>\u003Ctd>0\u003C\u002Ftd>\u003Ctd>0 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">5\u003C\u002Fth>\u003Ctd>0\u003C\u002Ftd>\u003Ctd>0\u003C\u002Ftd>\u003Ctd>0 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">7\u003C\u002Fth>\u003Ctd>+2\u003C\u002Ftd>\u003Ctd>4\u003C\u002Ftd>\u003Ctd>12.5 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">9\u003C\u002Fth>\u003Ctd>+4\u003C\u002Ftd>\u003Ctd>16\u003C\u002Ftd>\u003Ctd>50 %\u003C\u002Ftd>\u003C\u002Ftr>\u003Ctr>\u003Cth scope=\"row\">Total\u003C\u002Fth>\u003Ctd>0\u003C\u002Ftd>\u003Ctd>32\u003C\u002Ftd>\u003Ctd>100 %\u003C\u002Ftd>\u003C\u002Ftr>\u003C\u002Ftbody>\u003C\u002Ftable>\n\n\u003Ch2 id=\"sample-or-population\">Sample or population standard deviation, which one applies?\u003C\u002Fh2>\n\u003Cp>A teacher holding the marks of all 24 pupils in one class already has every number that matters, so the divisor is \u003Cstrong>n\u003C\u002Fstrong>. An inspector who measures 30 bolts out of a run of 4,000 holds a fragment, which calls for \u003Cstrong>n − 1\u003C\u002Fstrong>. The length of the list settles nothing here. A list of 8 numbers can be a sample and a list of 800 can be a population; the divisor follows how far the conclusion reaches.\u003C\u002Fp>\n\u003Cp>Most real data sets turn out to be samples, which is why n − 1 is the default here, as it is in Excel's STDEV.S and in the sd() function of R. That smaller divisor has a name worth knowing, \u003Cstrong>Bessel's correction\u003C\u002Fstrong>. A sample rarely contains the true extremes of the population it came from, so its raw squared distances understate the real spread; dividing by one less value pushes the answer back up.\u003C\u002Fp>\n\u003Cp>Picking the wrong divisor changes the number without usually changing the decision. On the eight default values the two answers sit at \u003Cstrong>2.13809\u003C\u002Fstrong> and \u003Cstrong>2\u003C\u002Fstrong>, close to 7 % apart. On the seven test scores 85, 92, 78, 88, 95, 71 and 90, the sample answer is \u003Cstrong>8.423324\u003C\u002Fstrong> against \u003Cstrong>7.798482\u003C\u002Fstrong> for the population, roughly 8 % apart. That gap narrows as the list grows.\u003C\u002Fp>\n\n\u003Ch2 id=\"meaning\">What does a standard deviation of 1.5 mean?\u003C\u002Fh2>\n\u003Cp>On its own, 1.5 means nothing until it is read against the mean it came from. Exam marks averaging 8 out of 20 with a standard deviation of 1.5 describe a group that spreads far enough to rank; a set averaging 150 with the same 1.5 describes values that are all but identical. As a proportion, 1.5 against a mean of 8 is \u003Cstrong>18.75 % of the mean\u003C\u002Fstrong>, the ratio known as the coefficient of variation.\u003C\u002Fp>\n\u003Cp>On data with a roughly bell-shaped distribution, close to \u003Cstrong>68 % of the values\u003C\u002Fstrong> fall within one standard deviation of the mean and about 95 % within two. A mean of 8 with a standard deviation of 1.5 therefore puts two thirds of the observations between \u003Cstrong>6.5 and 9.5\u003C\u002Fstrong>, almost all of them between 5 and 11. Skewed data with a long tail breaks that rule; the deviation table is where that shows.\u003C\u002Fp>\n\u003Cp>Two sets with the same mean can describe different situations. Delivery times averaging 30 minutes with a standard deviation of 3 minutes arrive when promised; the same average with a standard deviation of 15 minutes covers a band running from \u003Cstrong>15 to 45 minutes\u003C\u002Fstrong>.\u003C\u002Fp>\n\n\u003Ch2 id=\"edge-cases\">What is the standard deviation of a single value or of identical values?\u003C\u002Fh2>\n\u003Cp>Entering one number leaves the sample answer blank. With n = 1 the divisor n − 1 falls to zero, so the sample standard deviation is \u003Cstrong>undefined\u003C\u002Fstrong> instead of 0. The page prints that word in place of a figure. The population version returns \u003Cstrong>0\u003C\u002Fstrong> in the same case, since one point cannot be spread out.\u003C\u002Fp>\n\u003Cp>Five identical values behave differently. Every deviation is zero and the sum of squares is zero, so both answers come out at \u003Cstrong>0\u003C\u002Fstrong> while n = 5 still leaves a valid divisor of 4. Zero is a genuine result there; a \u003Cstrong>range of 0\u003C\u002Fstrong> between the minimum and the maximum confirms it at a glance.\u003C\u002Fp>\n\u003Cp>The page keeps \u003Cstrong>six decimals\u003C\u002Fstrong> and drops trailing zeros, which is why the sample answer on the default set reads 2.13809 while the population answer reads a bare 2. Reports rarely need that much, so \u003Ca href=\"\u002Fmath\u002Frounding-calculator\">round the figure\u003C\u002Fa> to the precision of the measurements behind it.\u003C\u002Fp>","2026-08-06T07:18:57.993Z","2026-08-28T10:00:01.764Z","2026-08-28T10:00:00.000Z","en","Standard Deviation Calculator",[16,20,24,28,32,36],{"id":17,"question":18,"answer":19},2406,"How do you calculate standard deviation?","\u003Col>\u003Cli>Add the values and divide by how many there are to get the mean.\u003C\u002Fli>\u003Cli>Subtract the mean from each value to get its deviation.\u003C\u002Fli>\u003Cli>Square every deviation.\u003C\u002Fli>\u003Cli>Add the squares together.\u003C\u002Fli>\u003Cli>Divide that sum by \u003Cstrong>n − 1\u003C\u002Fstrong> for a sample or by \u003Cstrong>n\u003C\u002Fstrong> for a population, then take the square root.\u003C\u002Fli>\u003C\u002Fol>\u003Cp>On 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squares add up to 32; the two divisors then give \u003Cstrong>s ≈ 2.13809\u003C\u002Fstrong> and \u003Cstrong>σ = 2\u003C\u002Fstrong>.\u003C\u002Fp>",{"id":21,"question":22,"answer":23},2407,"What is the standard deviation of 5, 9, 8, 12, 6, 10, 6, 8?","\u003Cp>Those eight numbers add up to 64 for a mean of \u003Cstrong>8\u003C\u002Fstrong>. The squared deviations total \u003Cstrong>Σ(x − x̄)² = 38\u003C\u002Fstrong>. The 12 alone accounts for 42.1 % of that.\u003C\u002Fp>\u003Cp>Read as a sample, 38 divided by 7 gives a variance of 5.428571 and a standard deviation of \u003Cstrong>s ≈ 2.329929\u003C\u002Fstrong>. Read as the whole population, 38 divided by 8 gives a variance of 4.75 and \u003Cstrong>σ ≈ 2.179449\u003C\u002Fstrong>. The values run from 5 to 12 for a range of 7.\u003C\u002Fp>",{"id":25,"question":26,"answer":27},2408,"How do I calculate standard deviation in Excel?","\u003Cp>Excel splits the two divisors across two functions. \u003Cstrong>=STDEV.S(A1:A8)\u003C\u002Fstrong> divides by n − 1 and matches the sample answer here. \u003Cstrong>=STDEV.P(A1:A8)\u003C\u002Fstrong> divides by n and matches the population answer. The older names STDEV and STDEVP still work and map to the same pair, sample first then population.\u003C\u002Fp>\u003Cp>Both families skip text and empty cells inside the range, so a column pasted out of a report may rest on fewer numbers than it has rows. The count on this page is the quickest cross-check.\u003C\u002Fp>",{"id":29,"question":30,"answer":31},2409,"Should I use the sample or the population standard deviation?","\u003Cp>Use \u003Cstrong>n − 1\u003C\u002Fstrong> whenever the numbers stand in for a larger group: a survey of 200 customers, one batch of parts out of a week of production. Use \u003Cstrong>n\u003C\u002Fstrong> only when the list is complete and the conclusion stops there, as with the salaries of all 40 employees of a company.\u003C\u002Fp>\u003Cp>When in doubt, n − 1 is the safer pick, since it returns the larger figure and never understates the spread. Both sit side by side on the page, so a wrong pick shows up as a gap between the two numbers.\u003C\u002Fp>",{"id":33,"question":34,"answer":35},2410,"Can a standard deviation be negative?","\u003Cp>No. Squaring every deviation wipes out the sign; the square root of a non-negative number is itself non-negative. The floor is \u003Cstrong>0\u003C\u002Fstrong>, reached only when every value is identical, as with a list of five 10s.\u003C\u002Fp>\u003Cp>Negative data causes no trouble. The set −8, −3, 0, 4, 12 and −5 has a mean of 0 and a sample standard deviation of \u003Cstrong>7.183314\u003C\u002Fstrong>.\u003C\u002Fp>",{"id":37,"question":38,"answer":39},2411,"What is the difference between variance and standard deviation?","\u003Cp>The variance is the average squared deviation and the standard deviation is its square root. On the default set the sample variance is \u003Cstrong>4.571429\u003C\u002Fstrong> while the sample standard deviation is \u003Cstrong>2.13809\u003C\u002Fstrong>.\u003C\u002Fp>\u003Cp>The units separate them in practice. A variance computed on weights in kilograms comes out in kilograms squared, a unit that matches nothing physical, while the standard deviation returns to kilograms and can be read next to the mean. Both figures appear here for each divisor.\u003C\u002Fp>",[],{"id":42,"documentId":43,"uid":44,"name":45,"tagline":46,"hubContent":47,"createdAt":48,"updatedAt":48,"publishedAt":49,"locale":13},15,"s8cujbpmiszotf6zdbotc2p0","math","Math","Calculators for school, work and everyday numbers","\u003Cp>Every calculator in this category computes live as you type and shows the formula behind the result. Grades, percentages, fractions, ratios or volumes: you see the answer and the reasoning, so you can trust the number you copy. Each tool also documents its edge cases, because a calculator you cannot verify is just a guess with confidence.\u003C\u002Fp>","2026-07-17T11:46:54.883Z","2026-07-17T12:01:48.544Z",{"id":51,"metaTitle":52,"metaDescription":53,"keywords":54,"metaRobots":54,"structuredData":54,"metaViewport":54,"canonicalURL":54},605,"Standard Deviation Calculator: Sample, Population, Variance","Standard deviation calculator that gives the sample (n − 1) and population (n) answers side by side, plus the variance, the mean and every deviation.",null,[],[57,69,81,93],{"id":58,"documentId":59,"slug":60,"term":61,"definition":62,"relatedTools":63,"createdAt":66,"updatedAt":67,"publishedAt":68,"locale":13},17,"o1ljpluv6app6z9rrhst4gfn","cubic-yard","Cubic yard","\u003Cp>A \u003Cstrong>cubic yard\u003C\u002Fstrong> is the volume of a cube measuring 3 x 3 x 3 feet, which equals 27 cubic feet or about \u003Cstrong>0.7646 m³\u003C\u002Fstrong>. It is the standard unit for ordering bulk materials in the United States: ready-mix concrete, topsoil, gravel, sand and mulch are all priced and delivered by the cubic yard, often shortened to \"yard\" on a quote.\u003C\u002Fp>\u003Cp>For scale, a typical 12 x 10 ft patio slab poured 4 inches thick contains 40 cubic feet, which is 40 \u002F 27 = 1.48 cubic yards of concrete. A standard ready-mix truck carries 8 to 10 cubic yards, so that whole slab uses less than a fifth of one load.\u003C\u002Fp>\u003Cp>How big is a cubic yard in practice? The cube stands 3 feet on each side, roughly counter height, and holds about 14 full wheelbarrow loads of material. Spread out, one cubic yard covers close to 100 square feet at a depth of 3 inches, the usual layer for mulch or a gravel path.\u003C\u002Fp>",[64,65],"math\u002Fconcrete-calculator","math\u002Fvolume-calculator","2026-07-18T14:23:56.812Z","2026-08-02T10:00:01.228Z","2026-08-02T10:00:00.000Z",{"id":70,"documentId":71,"slug":72,"term":73,"definition":74,"relatedTools":75,"createdAt":78,"updatedAt":79,"publishedAt":80,"locale":13},16,"mdw0ypy9f3z1nioxaxnl9rvr","password-entropy","Entropy (passwords)","\u003Cp>\u003Cstrong>Entropy\u003C\u002Fstrong> measures how unpredictable a password is, expressed in bits. Each additional bit doubles the number of guesses an attacker needs. For a randomly generated password, it follows a simple formula: entropy = length × log2 of the alphabet size, so both the length and the variety of characters raise the score.\u003C\u002Fp>\u003Cp>A random 16-character password drawn from the 94 printable ASCII symbols reaches 16 × log2(94), about \u003Cstrong>105 bits\u003C\u002Fstrong>. At 10 billion guesses per second, exhausting that space would take on the order of 10^14 years, while a 6-character password from the same alphabet peaks at 39 bits and falls in about a minute.\u003C\u002Fp>\u003Cp>Written as a formula, \u003Cstrong>E = L × log2(N)\u003C\u002Fstrong>: the entropy equals the length L multiplied by the base-2 logarithm of N, the number of possible symbols per character. Current guidance treats 75 to 80 bits as the comfortable minimum for accounts that matter, a level a random 12-character password mixing all character types already clears.\u003C\u002Fp>",[76,77],"generator\u002Fpassword-generator","generator\u002Fusername-generator","2026-07-18T14:23:55.755Z","2026-08-01T10:00:01.360Z","2026-08-01T10:00:00.000Z",{"id":42,"documentId":82,"slug":83,"term":84,"definition":85,"relatedTools":86,"createdAt":90,"updatedAt":91,"publishedAt":92,"locale":13},"ypkfs6wnfljm03ge2b1ztiwv","percentage-point","Percentage point","\u003Cp>A \u003Cstrong>percentage point\u003C\u002Fstrong> is the unit used to express the arithmetic difference between two percentages. It compares rates by subtraction, while \"percent\" compares them by division: the two measures answer different questions and can differ wildly for the same change.\u003C\u002Fp>\u003Cp>Example: an interest rate that moves from 5% to 7% rises by \u003Cstrong>2 percentage points\u003C\u002Fstrong> (7 - 5 = 2), but by 40 percent in relative terms (2 ÷ 5 × 100 = 40). Saying \"the rate went up 2%\" would be wrong on both counts: the correct phrasings are \"up 2 points\" or \"up 40%\".\u003C\u002Fp>\u003Cp>The standard abbreviation is \u003Cstrong>pp\u003C\u002Fstrong>, sometimes just \"points\", as in \"unemployment fell by 0.5 pp\". Confusing percent with percentage points remains the classic trap: a party moving from 20% to 30% in the polls gains 10 percentage points, yet grows by 50 percent, and headlines regularly pick the wrong figure.\u003C\u002Fp>",[87,88,89],"math\u002Fpercentage-calculator","math\u002Faverage-calculator","finance\u002Fcalcul-remise","2026-07-18T14:23:54.573Z","2026-07-31T10:00:01.940Z","2026-07-31T10:00:00.000Z",{"id":94,"documentId":95,"slug":96,"term":97,"definition":98,"relatedTools":99,"createdAt":102,"updatedAt":103,"publishedAt":104,"locale":13},14,"g3okb797ekrz0kkpm36mfjwc","weighted-average","Weighted average","\u003Cp>A \u003Cstrong>weighted average\u003C\u002Fstrong> is an average where each value counts in proportion to an assigned importance, called its weight, instead of counting equally. Each value is multiplied by its weight, the products are added, and the sum is divided by the total of the weights. Class grades, GPAs, and stock indexes are all weighted averages.\u003C\u002Fp>\u003Cp>Example: scores of 92, 85, and 88 with weights of 30, 30, and 40 give (92 x 30 + 85 x 30 + 88 x 40) \u002F 100 = \u003Cstrong>88.3\u003C\u002Fstrong>. The plain average of the same scores is 88.33, but if the heaviest score dropped to 78, the weighted average would fall to 84.3 while the plain average only fell to 85.\u003C\u002Fp>\u003Cp>The weighted average formula in plain words: multiply each value by its weight, add up the products, then divide by the sum of the weights. In a spreadsheet the whole calculation fits in one cell, since Excel's SUMPRODUCT function multiplies and adds in a single pass: SUMPRODUCT of the values and weights, divided by SUM of the weights.\u003C\u002Fp>",[88,100,101],"math\u002Fgrade-calculator","math\u002Fgpa-calculator","2026-07-18T14:23:53.329Z","2026-07-30T10:00:01.216Z","2026-07-30T10:00:00.000Z",{"slugs":106},[107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,6,160,161],"age-calculator","average-calculator","cd-calculator","concrete-calculator","cursive-font-generator","date-calculator","fantasy-name-generator","final-grade-calculator","fraction-calculator","glitch-text-generator","gpa-calculator","grade-calculator","hex-converter","hours-calculator","interest-calculator","military-time-converter","roman-numeral-converter","password-generator","binary-converter","celsius-to-fahrenheit-converter","kg-to-lbs-converter","percentage-calculator","word-counter","tip-calculator","morse-code-translator","username-generator","ratio-calculator","sales-tax-calculator","small-text-generator","mm-to-inches-converter","volume-calculator","timer","square-footage-calculator","stair-calculator","roof-pitch-calculator","board-foot-calculator","aspect-ratio-calculator","rounding-calculator","random-letter-generator","mulch-calculator","fence-calculator","upside-down-text-generator","acreage-calculator","gravel-calculator","proportion-calculator","paint-calculator","tile-calculator","braille-translator","quadratic-formula-calculator","gcf-calculator","probability-calculator","hex-to-rgb-converter","uuid-generator","lcm-calculator","caesar-cipher"]