How Many Values Are In The Range 35 To 95? Trust The Answer

Have you ever wondered how to spell ranges of numbers correctly? Should you use a hyphen, dash, or dash? How about whether to use spaces or not? And whether to use “from” and “to” instead of all those pesky hyphens? Don’t be surprised anymore.

Whether you write fiction or non-fiction, business proposals or blog posts, you’ll sooner or later face a similar dilemma as these beauties (answers at the bottom of the page):

Do I write “2000 – 2050”, “between 2000 – 2050” or “from 2000 – 2050” (Hint: it’s none of these.)

Is it “from £50 – £100”, “£50 – £100” or “between £50 and £100”? (Hint: two of these are correct)

Should I write “two to 10”, “2-10” or “from 2-10”? (Hint, one of these is correct)

I’ll go through some general punctuation, spacing, and grammar rules for ranges of numbers, then we’ll look at specific examples when it comes to ranges of years, temperatures, currencies, and more.

A quick note: I’ve used italics for correct examples and quotation marks for incorrect examples to avoid confusion.

General rules for number ranges

Hyphen, hyphen or hyphen?

When writing ranges of numbers, most style guides agree on using a hyphen (although the AP style guide prefers a hyphen). Don’t use a dash.

If you need a refresher on the difference between the three types of dashes, here it is:

Hyphen (-) The shortest of the three and the most commonly misused with spaces when a hyphen with no spaces is needed (unless the AP style is explicitly used).

The shortest of the three, and the most commonly misused with spaces when a hyphen without spaces is needed (unless AP style is explicitly used). En Strich (–) The champion of the hour. This is the little guy – or lass – you need for number ranges.

The champion of the hour. This is the little guy – or lass – you need for number ranges. Em Hyphen (—) The longest of the three dash types. It has its specific uses, but a range of numbers isn’t one of them.

How do I type a hyphen?

I get it. It’s one thing to know which stroke to use, it’s quite another to find that thing on your keyboard. The dash is simple, it’s always there whether on Mac or Windows, and faithfully shares real estate with the underscore.

To enter a dash:

Mac Users – Option + Hyphen (On your Mac, the Option key may be labeled Alt)

– Option + dash (on your Mac, the Option key may be labeled Alt) Windows users – More complex. The only method I’ve found to work on my Windows 10 laptop is through the emoji window. To bring up this popup, tap the Windows key + period/period key (.). Select the “Symbols” icon (third tab at the time of writing) and you will find the dash “en” on “General Punctuation”. Tab (denoted by ‘<'). Other methods that might work for you (they don't work for me, even with number lock on) are: Alt + 0150 on the numeric keypad (or with NumLock enabled), or ctrl + minus on the numeric keypad. For completeness, to get an em dash: Mac users press Option + Shift + dash; Windows users use the same instructions as for the dash if you're using the Emoji window, or Alt + 0151 if you have a separate numeric keypad (or if NumLock works for you), or Ctrl + Alt + Minus key. Distance In general, when working with number ranges, do not include a space on either side of the en dash. Keep things cozy by closing those gaps. No social distancing required: 1997-2001: Head chef and bottle washer at Gasthaus Berni. Before Friday, read chapters 7–9. It must have been 1-2 degrees yesterday. My fingers were numb! grammar There are three ways to express ranges of numbers, whether it's years, ages, temperatures, page numbers, and more: Use the hyphen as we saw, e.g. 1-10 years from/to, e.g. From 1 to 10 years Between/and, e.g. Between 1 and 10 years It is the mere existence of this choice that causes confusion. One of the most common mistakes I see when editing and proofreading is mixing two of the three methods listed above. The following examples are both incorrect: "I spent between £150 and £200." (Mixture of between/and option and use of dash en) "We must read from page 5-25 before next week." (Mixture of the "from/to" option and use of the dash en) If you decide to use one of these three methods, keep in mind that not all screen readers (used by people with visual impairments) can interpret when you are writing for the web or for a document to be read on the screen en hyphens . In this case, it's best to use words instead of hyphens (from ... to or between ... and). Don't mix numbers and words Many style guides (usually for nonfiction) recommend using words for numbers under 10 (two, seven, etc.) and digits for 10 and over (13, 145, 78, 585). That's all well and good, but what about a range of numbers that starts under 10 and ends over 10? consistency rules. In other words, use digits for the number under 10: there were 8-12 kids in the park. Elision in number ranges "Elision" in the context of number ranges is a fancy term for truncating the beginning of the second digit in the range. The following areas contain primed numbers: 41-2 156-7 1987-92 Style guides vary in their advice on eliminating number ranges. Below are the Chicago rules, with a table below comparing my example number ranges to Oxford style (AP style doesn't seem to specify an elision rule - if you know otherwise, drop me a note in the comments!): Use all digits (no omission) if at least one number is in the range 1 to 100, e.g. 2-10, 67-69, 82-323. For multiples of 100, use all digits, e.g. 100-107, 300-329, 2200-2254. If at least one number is in the range 101-109, 201-209, 301-309, etc., show only what has changed, e.g. 103-9, 308-64, 4.201-8. In all other cases, use two digits (unless more than two digits are required to indicate what has changed), e.g. 286-89, 593-678, 1053-56, 2869-4001. Don't omit roman numerals in number sequences; they are always spelled out, e.g. xxvi–xxvii. Actual number range Chicago Oxford 2-10 2-10 2-10 67-69 67-69 67-9 82-323 82-323 82-323 100-107 100-107 100-107 300-329 300-329 300-29 2,200 -2254 2200-2254 2200-54 103-109 103-9 103-9 308-364 308-64 308-64 4201-4208 4201-8 4201-8 286-289 286-89 286-9 5936-5 3-6936-8 678 1053-1056 1053-56 1053-6 2869-4001 2869-4001 2869-4001 Note that because of the way they are pronounced out loud, Oxford prefers to omit numbers as little as possible, except for teenagers (10 to 19). Check out my post on the best books for editors and proofreaders, including style guides range of years When writing a series of years, both the Chicago Manual of Style and Oxford advise the following when using a dash: If the years are within the same century and do not include a "00" year, omit as many digits as possible, e.g. 1201-67, 1939-45, 2001-9. If the two years given are from different centuries or contain the year "00", do not omit any digits, e.g. 1234-1387, 1800-1801, 1999-2009. However, if a title contains a range of two years in the same century, do not omit digits, e.g. The War of the Roses, 1455–1487. , do not leave out digits, e.g. The War of the Roses, 1455–1487. In addition, when indicating the dates on which a person lived, indicate all digits, e.g. Charles Dickens (1812-1870) The Oxford Style Guide adds that if the two dates are approximate then the c is repeated. Abbreviation for each date, e.g. c. 1345-c. 1578 (c. should be in italics followed by a space). The AP style isn't very accommodating in this regard, but my online dig suggests that AP prefers to repeat two digits for year ranges under 10 years, but you can also repeat all digits. For a range over 10 years, repeat all four digits. You can also use the "between... and..." and "from... to..." pairings for year ranges, but remember to only use one of these formats, not a hybrid: From 1990 to 1995 (not "From 1990–1995" and not "Between 1990–1995") Between 2000 and 2010 (not "From 2000-2010" and not "Between 2000-2010") BC and AD or BCE and CE BC/AD and BCE/CE are abbreviations used in the Gregorian calendar - the global standard for measuring dates. Either BC or BCE refer to years before Jesus was born; either AD or CE refer to years after Jesus' birth. BCE and CE are the secular versions of BC and AD, respectively: BC - before Christ AD – Anno Domini (in the year of the Lord) BCE – Before the common (or current) era CE - common (or current) era In your number range, if both dates are AD, there is no need to include the abbreviation "AD", although some authors prefer to include it if the year is before the first millennium for clarity: 1234–1387, 1800–1801, 1999–2009, 3–10 AD, 675–700 AD From 1234 to 1387, from 3 to 10 AD Between 1800 and 1801, between 3 to 10 AD If either or both dates are BC, you must specify BC/AD accordingly. While you could argue that it should be obvious when a date range is fully BC since the numbers count backwards, it's best to include BC for clarity. Some general rules from the Oxford Style Guide: Write AD/BC in small caps (not all style guides mandate this anymore, so it's worth checking the guide you're following. I didn't use small caps here.) BC comes after the year while AD comes before it (because it means "in the year of the Lord") (Chicago style also dictates this). Leave a space between the abbreviation and the year (also Chicago style) Year spans in BC are always written in full (not crossed out) as a crossed out date could be misread as a complete year (due to the years counting backwards) (not sure if this is Chicago style but it seems basic common sense to me) . !) Some examples: 20 B.C. - AD 100; from 20 B.C. to 100 AD 354 BC 436 B.C. BC; from 354 BC to 436 BC 187 BC - 125 BC (not shifted to 187 BC – 25 BC); from 187 BC to 125 BC When a date is approximate, spanning BC and AD, the Oxford Style Guide says to write it as follows: c. 17 B.C. - c. AD 78 (c. should be in italics followed by a space). When writing date ranges, the same rule applies as for ranges of other numbers. Use a dash with no spaces between the two dates (like Chicago style). NB. The following examples use the British date order day, month, year. The US order is month, day, year, and usually has a comma before the year, unlike the usual British English spelling of the date. Also NB. These date ranges are inclusive (the event that occurs within the date range ends on the second date, not the day before). Use the whole date (day, month, year) when the years are different, e.g. January 1, 1990–5. February 1995. Include the year after the second date only if the months are the same, e.g. January 1–5 February 1990. Only add the month and year after the second date if they are the same, e.g. 1st-5th January 1990. As with a range of years, you can also use the pairings "between ... and ..." and "from ... to ..." but, repeat after me, no hybrid versions mixing these with the hyphen format: From January 1, 1990 to February 5, 1995 (Also acceptable in the US is: From January 1, 1990 to February 5, 1995) Between January 1, 1990 and February 5, 1995 Years other than calendar years If you need to specify a range of years, e.g. For example, an academic year or a tax year, do not use a hyphen. The Oxford Style Guide states that a solidus or slash and elide years (even BC years) should be used: 78/7 BC tax year 2001/2 Academic year 1997/8 The AP style recommends using a dash instead of a hyphen, as a slash may not be transcribed correctly in some systems: 78-7 B.C. Chr Tax year 2009-10 Academic year 1997-8 The Chicago style sticks to the hyphen en, like any other eliminated range of numbers: 78-7 B.C. Chr Fiscal year 2009–10 Academic year 1997–8 Check out my post on the best books for editors and proofreaders, including style guides Number ranges for currency The first thing to note here is that most styles advocate repeating the currency symbol to avoid confusion. And yes, you guessed it, use a dash with no spaces: £100 - £120 $450-$455 Again, it's perfectly acceptable to use the pairings "between... and..." and "from... to...", but again with the feeling: don't cross-pollinate: Between £100 and £120 (not "Between £100-£120") From $450 to $455 (not "From $450 to $455") Don't use a mix of numbers and words, or a mix of currency symbol and currency word in the same area (or even throughout your document - strive for complete consistency). So don't write "From £100 to £120" (incorrect) or "Between $450 and $500 (incorrect). Finally, keep in mind possible misunderstandings when writing currency ranges with high numbers. For example, "From one to five million pounds" could mean either "£1-£5,000,000" or "£1,000,000 to £5,000,000". When in doubt, write out both numbers: from one million to five million pounds. Number ranges for page numbers The Chicago style follows the same rules that we have already seen for generic number sequences. In memory of: Use a dash (not a hyphen or dash) with no spaces on either side. Use all digits (no omission) if at least one number is in the range 1 to 100, e.g. 2-10, 67-69, 82-323. For multiples of 100, use all digits, e.g. 100-107, 300-329, 2200-2254. If at least one number is in the range 101-109, 201-209, 301-309, etc., show only what has changed, e.g. 103-9, 308-64, 4.201-8. In all other cases, use two digits (unless more than two digits are required to indicate what has changed), e.g. 286-89, 593-678, 1053-56, 2869-4001. Don't omit roman numerals in number sequences; they are always spelled out, e.g. xxvi–xxvii. Also, "p" or "pp" should only be used where it is not clear that the numbers are page numbers. This verified post on Chicago-style pagination has more details. The Oxford style for page number ranges also follows the guidance for generic number ranges. Refer to the table above which shows the differences between the Chicago and Oxford systems, but in short, the Oxford style advises: Use a dash (not a hyphen or dash) with no spaces on either side (like Chicago). Limit yourself to as few numbers as possible – i. H. only repeat digits where absolutely necessary, except for teenagers (10 to 19) due to the way they are pronounced out loud. Note that unlike Chicago, Oxford style tends to use "pp." for a date range, e.g. pp. 67-78. For more information, see this reviewed article on Oxford-style referencing. Number ranges for units of measure First, a bonus tip: most style guides agree that there should be a space between a number and its unit of measurement if the unit is a word or an abbreviation of that word (e.g. 12kg instead of 12kg). If a symbol is used, close the room (5 °C, 12′, 66%). However, if you're working from a style guide, check their preference, as they may dictate that the space be closed in all cases, or that a space be used before certain symbols (such as AP-style temperatures). Now let's move on to the number ranges for units of measure. If the unit is the full word or an abbreviation of that word (feet/feet, meter/m, kilo/kg, etc.), use the unit of measure only after the second number in the range: 90-100 degrees 20-25 feet 6-7kg However, if a symbol is used (%, ºC, ‘, etc.), use it after both numbers: 50%-55% 5.4'–5.5' Percentage ranges and temperature ranges are discussed in more detail below. Number ranges for percentages As with other number ranges, use a dash with no spaces for percent ranges. Most style guides seem to agree on repeating the percent sign in a range, e.g. 15-20%. You can also express a percentage range using the pairings "between... and..." and "from... to...". You know what I'm gonna say, right? No crossing between the different formats: From 15% to 20% (not "From 15%-20%) Between 15% and 20% (not "Between 15%-20%) The Chicago style recommends using the word "percent" (that would be "percent" in British English) in non-technical content and the symbol "%" in technical documentation. Always use Chicago-style digits when specifying percentages. Oxford style states that in all cases a digit followed by the symbol "%" should be used instead of spelling "percent". Number ranges for temperature Temperature range sticky areas are where to put spaces, if any, and whether or not to repeat the degree symbol and "C" or "F". Let's clear up the ambiguity: Most styles do not recommend a gap on either side of the dash, e.g. 17ºC - 20ºC. For styles that have a space between the number and the degree sign, the unit of measure (C or F) is used only after the second digit, e.g. 17-20ºC. Again, note that there is no space on either side of the hyphen. Also note that the AP style requires a space and no "circle" degree symbol before C or F. For styles that do not contain a space between the digit and the degree symbol (e.g. Chicago), the degree symbol and unit of measure are repeated, e.g. 17ºC - 20ºC. As with other number ranges, the pairings "between...and..." and "from...to..." can also be used for temperature ranges. And please do not use a mixture of different formats: From 17ºC to 20ºC (not “From 17ºC–20ºC”) Between 17°C and 20°C (not "between 17°C and 20°C") sub-zero temperatures When writing ranges of numbers that contain sub-zero temperatures, it's best to avoid the hyphen as it looks odd and can be confusing. Use "between ... and ..." or "from ... to ..." instead: Winter temperatures in Alaska range from ➖7ºC to ➖23ºC. Check out my post on the best books for editors and proofreaders, including style guides in summary As we have seen, there are not always clear universal rules for writing number sequences. If you take away nothing else from this post, remember: don't mix the hyphen format with any of the "word" pairings ("between/and" or "from/to"). Pick one and stick with it. Most style guides agree to use hyphens without spaces for ranges of numbers instead of spaces or dashes. However, if you don't already have a style guide to stick to, it's best to pick your favorite or the one most used for your area of ​​expertise and stick to it. I've tried to include as many number range examples in this article as possible, but if you have unanswered questions or spot an error in my post, don't be shy! Shout it out in the comments so I can either help you out or make things right. Now go and hand in some numbers! Answers to dilemmas at the beginning of the post

Finding the range of a function, algebraic – Expii

How can values ​​not be in range? Values ​​that are not in range are values ​​that are impossible given the full functional scope of the function. They can be beyond an asymptote or values ​​that the function simply skips.

Place your cursor where you want to use the name in this formula.

Enter the first letter of the name and select the name from the list that appears.

Or choose Formulas > Use in Formula and choose the name you want to use.

The domain of the expression is all real numbers, except when the expression is undefined. In this case there is no real number that makes the expression undefined.

parameter estimation

There are a number of population parameters of potential interest when estimating health outcomes (or “endpoints”). Many of the outcomes we are interested in estimating are either continuous or dichotomous variables, although there are other types covered in a later module. The parameters to be estimated depend not only on whether the endpoint is continuous or dichotomous, but also on the number of groups studied. Additionally, when comparing two groups, it is important to determine whether the groups are independent (e.g., males versus females) or dependent (ie, matched or paired, such as a before-after comparison). The table below summarizes parameters that may be important to estimate in health-related studies.

Estimated parameters Continuous variable Dichotomous variable A sample mean Proportion or rate, e.g. e.g. prevalence, cumulative incidence, incidence rate Two independent samples Difference in means Difference in proportions or rates, e.g. B. Risk Difference, Rate Difference, Risk Ratio, Likelihood Ratio, Attributable Proportion Two dependent matched samples mean difference

confidence intervals

There are two types of estimates for each population parameter: the point estimate and the confidence interval (CI) estimate. For both continuous variables (e.g. population mean) and dichotomous variables (e.g. proportion of population) one first calculates the point estimate from a sample. Remember that sample means and sample proportions are unbiased estimates of the corresponding population parameters.

For both continuous and dichotomous variables, the confidence interval (CI) estimate is a range of likely values ​​for the population parameter based on:

the point estimate, e.g. B. the sample mean

the investigator’s desired confidence level (most commonly 95%, but any level between 0-100% can be selected)

and the sample variability, or standard error of the point estimate.

In fact, a 95% confidence interval means that if we took 100 different samples and computed a 95% confidence interval for each sample, then about 95 of the 100 confidence intervals would contain the true mean (μ). In practice, however, we select a random sample and generate a confidence interval that may or may not contain the true mean. The observed interval may overestimate or underestimate μ. Hence, the 95% CI is the likely range of the true, unknown parameter. The confidence interval does not reflect the variability of the unknown parameter. Rather, it reflects the amount of random error in the sample and provides a range of values ​​likely to contain the unknown parameter. Another way of thinking about a confidence interval is that it is the range of likely values ​​of the parameter (defined as point estimate + margin of error) with a given confidence level (similar to a probability).

Suppose we want to generate a 95% confidence interval estimate for an unknown population mean. This means that there is a 95% chance that the confidence interval contains the true population mean. Thus, P([sample mean] – margin of error < μ < [sample mean] + margin of error) = 0.95. The Central Limit Theorem, introduced in the Probability module, states that for large samples, the distribution of sample means is approximately normally distributed with a mean: and a standard deviation (also called standard error): For the standard normal distribution, P(-1.96 < Z < 1.96) = 0.95, i.e. H. there is a 95% chance that a standard normal variable Z will fall between -1.96 and 1.96. The central limit theorem states that for large samples: By replacing the expression on the right side of the equation: Using algebra, we can rearrange this inequality so that the mean (μ) is the middle term, as shown below. then and finally This last expression then gives the 95% confidence interval for the population mean, and this can also be expressed as: Thus, the margin of error is 1.96 times the standard error (the standard deviation of the point estimate from the sample), and 1.96 reflects the fact that a 95% confidence level was chosen. So the general form of a confidence interval is: Point Estimate + Z SE (Point Estimate) where Z is the value from the standard normal distribution for the selected confidence level (e.g. for a 95% confidence level Z = 1.96). In practice, we often do not know the value of the population standard deviation (σ). However, if the sample size is large (n > 30), the sample standard deviations can be used to estimate the population standard deviation.

Table – Z-scores for commonly used confidence intervals

Desired Confidence Interval Z-score 90% 95% 99% 1.645 1.96 2.576

A 95% confidence interval is most commonly used in the health-related publications, but this is an arbitrary value and other confidence levels can be selected. Note that the 99% confidence interval for a given sample would be wider than the 95% confidence interval because there is more confidence that the unknown population parameter is in the interval.

Confidence interval estimates for smaller samples

For smaller samples (n < 30), the central limit theorem does not apply, and another distribution called the t-distribution must be used. The t-distribution is similar to the standard normal distribution, but takes on a slightly different shape depending on the sample size. In a way, the t-distribution could be thought of as a family of distributions for smaller samples. Instead of "z" values, there are "t" values ​​for confidence intervals that are larger for smaller samples, resulting in larger margins of error because small samples are less accurate. t values ​​are listed by degrees of freedom (df). As with large samples, the t-distribution assumes that the result of interest is approximately normally distributed. A table of t values ​​is shown in the frame below. Note that the table can also be accessed from "Other Resources" on the right side of the page. back to top | previous page | Next page