foreign so good morning everyone so good day to all of you today we're going to have our second lecture in General Physics one so to continue our discussion on the introductory topics in physics we are going to highlight how to estimate errors and uncertainties in measurements our objective for this lecture or the following so to differentiate accuracy from precision and to evaluate the accuracy and precision of measurements using um by estimating the errors and uncertainties of our measurement so the flow of this lecture is um will consist of concept Builder so once we got into our concept Builders you may pause this video and resume watching after you submitted your answers in the given pundit link and a brief discussion of the concepts so the quality of our laboratory experiments depends on the quality of the data measured so how successful is our laboratory experiment is similar to asking how accurate or how precise our data set is so we will begin our discussion by defining accuracy and precision so to do that let us have a simple activity so your explanation here you will you can submit it in our in the given palette link so I would like you to copy and draw this circle in your notebook so copy as much as possible and then afterwards I would like it to repeat it three more times so draw the circles three more times the way using this activity how are you going to Define and differentiate accuracy and precision so in this context using this simple activity defines accuracy from Precision accuracy and precision so you may pause this video so in this context accuracy is how similar will be your drawn Circle to your reference Circle so if we have here our reference reference nothing so how similar are the size for example or the colors drawing circle more or black ink or other color young shape so perfect circle then reference so accuracy is the measure of how similar is your drone Circle to your reference Circle so in this context while precision is how similar are your drone circles to each other so if we have here our reference for example drawing more with or with no respect with with or without respect to your reference Circle we can measure the Precision of your drawing so in this context Precision is how similar are your drawn circles to each other with or with no due respect to the reference because measurements can be precise but inaccurate device opening precisive measurements are reference more so that's why we define precision here as um the measure of how similar are your drawn circles to each other circles so if not so for example um like for example in this illustration we can say that there is a problem in the quality of your work so in in the quality of your or in the ability in your ability to replicate the circle so for example again authentic Circle so for example you are to replicate Mona Lisa so your ability to copy Mona Lisa to the smallest details for example your environment that is the measure of your accuracy while your ability to replicate Mona Lisa multiple times is your level of precision that describes your Precision your ability mode to reproduce the same results over and over again so the idea is the same when we put it in the context of measurements that the quality of our laboratory experiments depends on the quality of the data measured so how successful is our laboratory experiment is similar to asking how accurate or how precise our data set is precision and accuracy are ways of describing the quality of your measurements and the best quality wait Lang okay so the best quality best result nothing is when your measurements are both accurate and precise so that is the best result the results if you want to to perform an experiment our best result is to have an accurate both accurate and precise result or data so here is a comparison between accuracy and precision so accuracy is defined as the degree of closeness I answer it is the degree of closeness or the measure of how close a measurement is to the True Value so improve value not in detail is also called as the actual the real the accepted or the theoretical value value or your experimental value to the true value is the measure of accuracy and on the other hand in Precision it is defined as the degree of closeness of the measurements to each other so it is the exactness the consistency or repeatability of measurement so Precision is how similar your measurements are simple activity not and precision is how similar are your drone circles to each other so here Precision is how similar are your measurements so measurements take note that measurements are precise when you measure the same item multiple times and the values are close to each other so however take note then plus that Precision tells you nothing about um the accuracy of your of your data set so this means that it is possible to be very precise but not accurate and it is also possible to be accurate without being precise because Precision is independent of accuracy so there are times now young measurements can be can be close to each other but far from the True Value so that is why um implication nothing that it is independent from accuracy and we cannot we cannot say that precisely measurements nothing accurate so so sometimes it's true and sometimes it's not true so that's why Precision is independent of from accuracy thank you so for a concept Builder number one so I would like you to describe the level of the accuracy and level of precision of the following Target and include also um a brief explanation come back at Union description in here so just use um high and low so just use high and low as your adjective for the level of precision so high and low engine high accuracy High Precision high accuracy low Precision High Precision low accuracy and low precision and low accuracy and then explanation so to answer this problem so our first Target here this is a dart boards is a target nothing represents our um Target value or our actual value so since you measurement nothing or you got nothing to Mama's eye so we can say that that this um this target depicts height accuracy and high Precision so therefore since close to each other it it has high precision and since the the the the values or the results are found at the bullseye then it also has high accuracy so our second target has low accuracy if this is our actual value result value but if we look at the the data set so if we look at the results in result nothing is close to each other that measurements can be very precise pero inaccurate and sometimes spreading accurate precise this this target target number two illustrates low accuracy error High precision for example this is a dart s eye however Target nothing Precision results and the last depict um low accuracy and low Precision so although malayo um and also the the results are far from each other that's why this target um illustrates slow accuracy and flow precision foreign currency and precision leads to potential errors in measurement so where we Define error as the difference between the measured value and the True Value so there are three General types of errors in measurement so these are blunders we have also systematic errors and random errors so let's start with blunders blunders are also known as personal errors so these are errors caused by carelessness and accidents that's why sometimes blunders is also called as accidental error so it is also called as accidental error or gross error or indica error so blunder is usually result to anomalous data in your experiment so when we say anomalous data this is a data that can extremely affect your data set during the analysis for example so for example um when you have to do multiple trials in an experiment that involves measuring 12 milliliter of a solution for example um you have to perform five trials then the first four trials most young results are 12.1 12.2 12.0 11.9 and then because of an accidental spill young last record or in a measure is just 10.5 ml so compared to this um first four set of your measured obtained value this is an anomalous this can be treated as an anomalous data so from when you when you have an anomalous data this data or the anomalous data is omitted from the analysis of the result because this can affect the quality of your data set another example would be carelessness in recording your um your measurements for example solito a scale might read 21 centimeter but because you are careless in recording your your sourdito your measured value you wrote down 12 CM instead of 21 so this will of course affect the quality of your data set the second type of errors in measurement is the systematic error so these are errors in procedures so meaning so there is an error in the process of obtaining your measurements or obtaining your data so systematic errors has unidentifiable cost and you can trace it back in the procedure of your experiments systematic error you can trace back the cost of your errors in the procedure or in the process in the design of your experiment so errors of this type result in measured values that are constantly too high or constantly too low compared to the expected or True Value so that is why when you commit systematic errors or when you have systematic errors in your measured value um it resulted to a precise so the measurements are precise but inaccurate yeah so if you have this type of error you need to go back to your design you need to go back to your procedure experimental procedure in order to troubleshoot this kind of error so the sources of systematic errors can be environmental observational instrumental or theoretical and a common type of systematic error is the offset offset or zero error so you commit this when when um you have problems with your instruments for example for example of an offset error is that you use broken ruler instead of a complete ruler so of course object so of course instead of getting the true value or getting a close value accurate value and another would be for the zero error is that sometimes ruler or other for example weighing scale Hindi calibrated for example weighing scale means zero but sometimes young Wings 105 or 100 or something correct measurements the same with the rulers um markings 4-0 but others so this this area here will give you an error in your measurement so which implies that you are already few millimeters away from the true measurement of your of the length of your measured of your object being measured so instead of getting nine four examples for example eight measure data and the last type of errors in measurement is the random error So Random errors are naturally occurring errors so these are errors due to unavoidable environmental conditions so most of the time the sources of this error cannot always be identified so since the error happens at a random it becomes difficult to trace the course of error unlike in the systematic error where you can always examine the the experimental procedure or the decide in order to troubleshoot your your error in measurement it's a random errors [Music] so when you commit this type of error your measurements um are imprecise so this affects Precision of your measurement so again systematic error your measurement new is precise but inaccurate random errors it affects the Precision accurate so some of your data may be accurate but some of your data may be inaccurate is not precise so this will result to a wide um difference in your um set of data variation in data so the sources of systematic errors can be environmental or observational so one type of random error is the reading error so which is which course is observational So reading or a liba foreign errors So reading error is an error that arises when um we do direct measurement of some quantities for example for example you are going to measure the length of this wood for example and then rulers so 3D for example so you need to estimate for example if this is 11 and this is 10. so so in this case um how well would you be able to provide the estimate so you need to learn how to provide the estimates when measuring a data because this will read um this will lead to a random error so accuracy is the measure of how small is the systematic error of an experiment so the smaller the systematic errors the more accurate are your measurements were while Precision is the measure of how small is the random error of an experiment so the lesser the variation in data the smaller are the random errors thus the more precise are your measurements so how do we now evaluate the accuracy and precision of your measurement so recall that accuracy is the correctness of your measured value in comparison to the True Value while um Precision is [Music] um the consistency of your measured values or how similar are your measured values or experimental values to each other so in evaluating the accuracy we need to solve for the error in our measurements and by solving for the error so solving for the error allows us also to evaluate the Precision of the measured values so measurement error is the difference between a measured quantity and its true value so it is the measure of how far away is your obtained measurement from the true or actual value of the measurement so there are different ways on estimating errors or for solving for the errors in our measurement so we can um solve for the absolute error percent error percent difference and we can also look for the mean variance and standard deviation in order to estimate the errors in our measurements so let us first Define each of the following so absolute error or Delta e is the difference between your obtained or experimental measurement and accepted measurement so here is our um formula for the absolute error so your absolute error nothing is the absolute of the difference between the experimental value or your young value measure and the true value or the accepted value so absolute it is simply the errors for example in error nothing is the experimental value minus the accepted values error nothing can be positive or negative and this sign symbolizes the notes whether you obtain value more is larger than the true value or you obtained volume is um lower the and the True Value so later we will have an example to differentiate error and absolute error so absolute error is the measure of how far is your obtained measurement to the accepted value so when solving for the absolute error you are measuring the actual size or the actual difference between your obtained and True Value so if we have here um 12 for example 12 is our um accepted value or True Value and then measured in the 13. so you you you absolute error simply just tells you between these two so what is the absolute error so absolute error will always be positive because of this absolute sign here so next we have percent error so percent error is the absolute error divided by the accepted value multiplied by 100 so here is our formula so percent error is equal to the absolute errors this is just this one so absolute error so absolute error then no need to solve for the absolute error you can you can um look for the absolute error before solving for the percent error so this equation above is simply the formula for the absolute error so percent error is the absolute error divided by the the accepted value times 100 so here we are relating we are relating the the error to our um to our true value so is the accepted or expected value or the theoretical values percent error measures how large is the error committed compared to the True Value it measures how large [Music] is the error the error compared to [Music] the true value so that is the percent error so it measures how big of a part is your error relative to the True Value and it tells you how big is the error of your measurements or how close is your measurement or your experimental value to the True Value foreign difference or percentage difference is the difference between two measurements divided by the average of the two measurements multiplied by 100 so here we have here x sub 1 and x sub 2 over in x sub 1 is um the first experimental value and X2 is the second experimental value so this this x of 1 and x sub 2 here are experimental values and then to get the percent difference we get the the absolute of the difference between two experimental values divided by their their average times 100 so um percent difference Compares um experimental values to find out how much these values differ from each other so it tells you how large or or small is your obtained data from other values in your data set so percent difference since it Compares um your your your obtained value from other obtained value percent difference indicates precision yeah so it indicates Precision of measurements since it takes all the experimental values and compare them to each other so again nothing percent difference it tells you how large or small is your experimental value together compared to other experimental value that is for the percent difference so let us have an example so here so three trials were performed to determine the guidance the distance rather between building a and Building B so the first group measured it to be 503.5 meters the second group measured it to be 502.8 meters while the third group measured it to be um 497.4 meters if the actual distance between building a and Building B is 500 meters so determine a the absolute error of the measurements B the percent error of the measurements and see so c um percent difference between two measurements so let's answer letter A so we would like to find the absolute error of the measurement so to look for the absolute error we simply use this equation so absolute error or Delta e is equal to the absolute of the difference between the expected and the accepted presentation so here we have here three trials conducted by group one group two and group number three and here are our experimental values so here it is this is our X of two this is our X of three so yeah so 503.5 500 2.8 and 497.4 so first we need to look for the error so yes experimental value minus the accepted value error can be positive or negative three positive 3.5 positive 2.8 and it due to negative 2.6 positive and negative so this means it's larger than your true value so 500 3.5 is larger than 500 meters so larger that by what number so it is larger by 3.5 meter so 500 2.8 is larger than 500 meters by 2.8 meters 2.8 meters while 497.4 is um smaller than 500 so smaller Channel 2.6 meters so to get the absolute error just simply get the absolute of this one absolute of course you will get the absolute error so again the positive and negative sign denotes the experimental value is higher than or lower than the actual value by the measure of error so absolute error measures the number of deviations from the actual value but will not tell you whether your obtained value is smaller or larger so absolute error absolute error whether you're this error tells you na you obtained value mobile is larger than the actual value or smaller than the Apple value signage you can tell whether so actual value okay so now that we have the percent error and the absolute error we can now look for the percent error so again percent error is equal to the absolute value or absolute error divided by the accepted um value times 100 so you just plug in this value substitute it to the equation so we're in an accepted value nothing is equal to 500 meters so if you plug all these values to this equation now um number one episode one which is 503.5 meters with an error of 3.5 meters so absolute around 3.5 meters 3.5 meters is 0.7 percent of 500 so 2.8 meters is 0.56 percent of 500 so 2.6 meters 0.52 of 500 so the percent error tell us um about tell us um how big is the error compared to the True Values for example the 3.5 is 0.7 percent of 500 so solving for the percent error will immediately give us hint on whether we need to worry or not about the errors in our data so okay so let us look at the following examples I have here so in this situation so the absolute errors of the two foreign so the absolute errors uh in the two measurements are the same so your absolute error nothing so it's a situation those situation one is 3.5 greater than 3.5 foreign Again by using the formula and then substituting the values detail given an absolute error so no need for this one so just change in E over x sub a times 100. so 3.5 divided by 500 times 100 is 0.7 percent situation to 3.5 divided by 20. 3.5 divided by 20 times 100 is equal to 17.5 percent error tells us how big is the error in our measurement relative to the True Value so young 3.5 error volume is smaller young 3.5 becomes a big part of your true value um error so for letter C so solving for the percent difference of any two measurements is measurement one and then measurement number two so between for example X1 and then X3 so between these two or patterning X2 and then X3 so any um between any um two um obtained measurement or experimental values in your data set so again this is our uh and towel data formula four percent difference X1 and then x sub 2 We're In It obtained values so by substituting it to our equation value so how big is the difference between the first and the second measurement so there is a zero point um 1391 percent difference between the first and second measurement or about um 0.14 percent if you round this up to two decimal places so the difference between the two measurements is 0.14 so this tells us that um that 503.5 is 0. uh zero point 14 bigger than 502. so for example is that 503 or x sub 1 now is 0.14 bigger done or larger than x of 2 or rather than x of 2 is 0.14 smaller than x sub 1 and you can also compare others for example my x sub data you can also compare it to any of these experimental values so take note percent difference is um tells you how large or how small is an experimental value compared to other experimental value that's why it is a measure of precision so let us have a concept Builder so you may pause this video and then um resume watching after you submitted your answers in the given padlet link so problem number one so Advanced object is dropped a distance of 100 meters and the acceleration of the object is found to be 10.2 meters per second squared so if the acceleration of a falling object is said to have the acceleration due to gravity which is 9.8 meter per second squared how large is your measured acceleration compared to the acceleration due to gravity and how large is your error how large is your error number two so two trials were performed in an experiment to determine the latent heat of vaporization else of V of water at 100 degrees Celsius so the values of of the latent heat of vaporization of water obtained were 532 calories per gram and 536 calories per gram so find the percent difference between the two values so you would like to know how much larger or smaller are these are these two values compared to each other okay so you may pause this video and just do some watching afterwards so to answer number one so we were asked how large is our measured acceleration compared to the acceleration due to gravity so you've measured acceleration nothing is 10 point um 10.2 meter per second squared so 10.2 um 10.2 meter per second squared compared to 9.8 meter per second squared so 10.2 meter per second squared is 0.4 meter per second squared larger than 9.8 meter per second squared so this is our um absolute error it just tells us the difference between the two values so between the the experimental and the absolute values so how large no man is the error so if this is our error how large is this error compared to our true value so to to solve for to answer that question we need to solve for the percent error so here so using this equation substitute with the given and we'll get a value of 4.1 percent so 0.4 percent or or yeah 0.4 percent or 0.4 meter per second squared is 4.1 percent all of 9.8 meter per second squared so this is how large is our air Works 4.1 percent of the True Value [Music] okay so to answer number two so we would like to know um we would like to compare two latent heat latent heat of vaporization of water so using percent difference equation at 10 substitute so we'll get this value so the percent difference between this measurement measured values is about 0.749 or 0.75 percent so meaning 536 is um 536 is 0.75 percent bigger than 532 calories per gram or 532 calories per gram is 0.75 percent smaller than 536 or lesser than 1026. [Music] foreign experimental values in our uh in our measurements so how do we solve for the accuracy of your data so in this case we sold for the mean of the data and then compare it to the actual value so if we have a data set or multiple values we get the mean and then compare this mean to your actual value say for example we have two sets data sets here so set a and set B and that the true value is 10 inches so how do we now solve for the accuracy we need to solve for the mean we need to get the mean in order to compare it to our um actual values so by the way we can also compare this individually but but if we want to know whether you're set of data is accurate or not we need to solve for the mean so we need to solve for the mean of set a and the mean of data set B so we don't know yet that values these values so to solve for the mean the formula is mean is equal to the summation of X where X is our individual um individual observation so we have here five observations so we get the sum of this five observations and then divided by the number of observations so n here so is equal to the number of observations we have here one two three four five observations for both set a and set B so in N nothing is equal to five so if we um get the sum of set a and then divided by five then we'll get this value so the mean of data set a is equal to 10.02 inches while for the mean of set B is equal to 8.08 inches so just by looking at this values just by looking at the mean so we can already infer that um set a is more accurate compared to set B Because actual value basis to to see whether a data set is accurate or not so just by looking at the mean and comparing it to our actual value here 10.02 10 inches compared to 8.08 inches so that's why that set a is more accurate but if you want to quantitatively describe the accuracy of your data set you need to solve for the error of your measurement so you can solve for the absolute error or you can solve for the percent error so in this case let's just use um the percent error in order to relate our um absolute error to relate our error to the actual value so percent error again it is equal to the absolute at or divided by the Apple value times 100 so young absolute error nothing is sold through this um equation so the absolute value is equal to the absolute of experimental value minus accepted value so your experimental value not 22 is not an individual value it is a set of of experimental values instead of x sub e foreign of data set a minus the actual value accepted divided by the actual value times 100. we have here 10.02 minus 10 divided by 10 and then times 100. so young percent error yeah the data set a nothing is equal to 0.2 percent while the percent error of data set B is equal to 19.2 percent so if you substitute again the values coming from data set B don't setting equations a formula percentage of a which is equal to 10.02 inches and mean of B which is equal to 8.08 inches we claim that um just by inspection inspecting the the mean and comparing it to the actual value infertile that said they is more accurate compared to set B pero to support your claim you need a value to support your claims you need to solve for the error of your um data set data set a is at least 19.2 percent errors a calculations is more accurate than data set B because just 0.2 compared to 19.2 percent yeah foreign we need to do multiple trials in order to collect a set of data so in this manner we would be able to minimize the error in our measurements set of data in order to to look for the the most probable value that would represent your data and that most probable value could be the value of your true value so when we don't know the True Value we consider the mean of the set of data the set of measurements as the most probable value so closest to the true values mean because we consider the mean as the most probable value meaning so this is the value that is closest to the True Value given a set of data is independent of accuracy but although this is the case we settle on how precise are the obtained values in order to accept that that the set of values can be used to provide a correct measures most of the time or there are times and hope that that these values can be can represent the true value or can be used to to find the True Value you need to include also uncertainty so that's why when the value is I know when the actual value is unknown we so we can solve for the mean variance and standard deviation and we need to include the uncertainties in your measurements so again mean is the average of The observed value so this is considered as the most probable value or just npv yeah most probable value so meaning it is considered as the most probable value or the value closest to the True Value so to to solve for the mean we use this formula so X bar is equal to the summation of X where X is our individual um individual observation so we need to get the sum of your individual observation then divide it by the number of observation so next is the variance so variance measures variability from the average or mean so it is the calculated it is calculated see by taking the summation of the squared difference between measurement and the mean parenthesis this is also called as the mean deviation so meaning individual measurement of your data set this is the mean deviation and this is our squared of mean deviation so therefore we need to get the squared of mean deviation and then we need to get the sum of this squared mean deviation divided by the number of observation in order for us to get the value of our variance so variance is the overall spread of data of your data next is the standard deviation the standard deviation is the square root of variance so if if the variance is already given in order to get the standard deviation you just need to get the square root of the variance if not you can proceed with this method so yeah equation so it is a measure of how diverse or Spread spread out are the measurements from their average the standard deviation equation again so Sigma is the standard division X of Y is the measurement of the height experimental value or the individual individual measurement and then um X bar is the mean or the average and N is the number of observations so for instance standard deviation is the measure so it's the measure of how scattered are your data set so variance since that is the case it is a variance in standard deviations or indicators opposition so it measures how far away the numbers from the average and how far away from each other are the measurements within a data set so a variance of zero and variance though nothing is zero meaning identical values measured values for example 12 centimeters measurements must be 12 centimeter also yeah larger variance means uh must widely spread must scattering data and if you have small variance it indicates that the values are close to one another which means they are precise which means precise values on the other hand the small standard deviation means that the most that most of the measurements are close to their average and a large standard deviation means that the the measurements are very diverse and they must they must be far away or farther than farther from the mean of your data set so if we have here the mean if you have a number a number line and this is the mean standard deviation standard deviation so for example one standard deviation nothing standard deviation mode um so one two so this is two standard deviations okay this is one standard deviation away standard deviation or for example 0.5 block so one standard deviation away so it was 0.5 language so two standard deviation away so it must have been another 0.5 so compared detox two standard deviation away is already two again although two standard deviation away so just recall your statistics so the standard deviation is an indicator of precision standard deviation can also be used as an expression of our error in measurements standard deviation to express the errors in measurements so therefore it can be used to to identify the accuracy of your measured values so let's see how measurements can be is reported so in reporting the measurement X of a physical quantity when given a set of data we use this expression so X is our measurement and X bar is the mean so plus or minus the standard deviation so this expression indicates that the best estimate of the true value is found between the mean plus or minus the standard deviation so for example so for example 10 is equal to 10 tapos and standard deviation nothing is equal to 0.5 that the True Value lies between so little X minus or mean minus standard deviation to mean plus standard deviation so 10 minus 9 0.5 is 9.5 and then 10 I know plus 0.5 is 10.5 so meaning that the true value or your actual value is found within within from 9.5 between 9.52 10.5 minutes your measurement is accurate as long as it falls within the range of values where the True Value lies or or it is more accurate if it is closer to this range of value so if you for example if you if you measure a physical quantity the same physical quantity equals 11. so given that the range is only from 9.5 to 10.5 so 11 is a not so accurate um value so foreign is found within this range of values activities so let us have a concept Builder so you may pause this video and just resume watching after you submitted your answer so problem solving so the following values were determined in a series of tape measurements of a line so 1000.58 1000.40 1000.38 1000.40 and 1000.46 meters so determine the following so a variance and variance of the measurements B standard deviation and see the measure of the line so it is mean of your data set so let's answer this problem so in solving a problem involving a set of data it is easier to um use a tabular if presented in tabular form so we have here our individual measurements X so x sub 1 x sub 2 x sub 3 x sub 4 and X applied so therefore a number of observation at n is five so first let's look for the mean of our what do we call this of our data set so again to get the mean so we use this formula observation and divided by the number of observation therefore you mean nothing is 1000.44 and then we need to solve and standard deviation we need to solve for the mean deviation so to get that we need to subtract our mean from our individual observation so 1000.58 minus 10.44 will give us 0.14 and repeating the procedure these are the gathered data so result noted from solving for the mean deviation so 0.14 negative 0.04 negative 0.06 negative 0.04 and 0.02 and then sold for the squared mean deviation so is wait Lang Okay so to get the the squared mean deviation just simply Square these values yeah so 0.14 squared is equal to 0.0196 and so on so if you recall the the formula for getting the variance is equal to this the is equal to the summation of the squared mean deviation so x minus X bar foreign so therefore we need to get the sum of this squared mean deviation and then divided by five so I'm result not in John will be again will be 0.00536 so that is our um variance and then new standard deviation so you can standard deviation nothing it's just the square root of variance so we have here 0.0732 so unique standard deviation nothing how do we now report this measurement is equal to 1000.44 so this is our most probable value so the central value that represents the data set so this is our most probable value what could be our true value and then we have an uncertainty so you standard deviation nothing it gives us the uncertainty in our measurement 1000.44 is the true value so therefore we need to express that doubt in order to express the error in our in our um estimate the error in our measurement so young uncertainty is our standard deviation so your standard deviation attended 0.07 so the standard deviation is the expression of our error since we do not know the actual value we suppose that the mean is the most probable value but since there is a doubt in our mean we include the measure of stud of uncertainty which is the standard deviation so therefore the True Value lies between 1000.44 plus or minus 0.077 meter so take note plus assume uncertainty nothing in Express launches a single significant digit so we need to round this to a single significant digit number that is for the uncertainty however for example 0.01 number so here so we can say so 10 1000.44 minus 0.707 so we have here one thousand 0.37 and then 1000.44 plus 0.07 equals 1000.51 so meaning the true value is found within this range of values so for the new value nothing is 1000.37 1438 up to 1000.51 so that is how we report our uh measurement together with its uncertainty foreign