Name: graphs, curve fit, and standard deviation
Uploaded: 2020-03-26T07:22:24.000Z
Duration: 4 min 34 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Chemistry

CHEM.1

CHEM.1.D

Scientific Measurement and Lab Skills

Traduzir descobertas experimentais de pontos de dados em representações visuais, como gráficos, é essencial para determinar a relação entre duas ou mais propriedades. Essas propriedades são chamadas de variáveis. Quando há duas variáveis, um gráfico criado a partir dos dados é chamado de bidimensional. O gráfico tem dois eixos. A variável independente é plotada no eixo x e a variável dependente é plotada no eixo y.
Tomemos, por exemplo, esses dados de amostra para a temperatura e o volume de um gás. O volume do gás depende da temperatura. Assim, traçaríamos a temperatura medida no eixo x e o volume no eixo y.
Quando há vários pontos de dados com o mesmo valor x, como se medirmos o volume várias vezes em uma temperatura, também calculamos o desvio padrão dessas medições. O desvio padrão é um valor estatístico, que indica a quantidade de variação presente em um conjunto de valores.
O desvio padrão é calculado usando esta fórmula, onde n é o número de pontos de dados, x barra é o valor médio dos pontos de dados e xi representa cada ponto de dados individual. Você pode calcular o desvio padrão manualmente ou um programa de planilha pode calculá-lo automaticamente. Quanto mais próximo o desvio padrão estiver de 0, mais próximos os pontos de dados estarão do valor médio. Se o desvio padrão for igual a 0, todos os pontos de dados inseridos terão o mesmo valor.
Vejamos os valores médios e desvios padrão das medições de volume em cada temperatura em nosso conjunto de dados. Podemos resumir cada conjunto de pontos de dados como a média mais ou menos o desvio padrão. Determinamos os algarismos significativos para cada média a partir das medições correspondentes e arredondamos os valores médios de acordo.
O desvio padrão de cada grupo deve ter o mesmo número de casas decimais que a média, então arredondamos cada desvio padrão para a casa dos centésimos. Para determinar graficamente a relação entre duas variáveis, podemos ajustar os dados com uma função de melhor ajuste.
A função é gerada automaticamente pelo software de planilha e pode assumir a forma de uma linha de tendência linear, uma função polinomial ou uma função exponencial ou logarítmica. No caso de nossos dados de temperatura e volume, a relação é linear. Portanto, os pontos de dados são ajustados por regressão linear de mínimos quadrados. Seu programa de planilha retornará a equação para a linha de melhor ajuste e um valor de r-quadrado. Quanto mais próximo o valor de r-quadrado estiver de 1, melhor será o ajuste dos dados.
Em seguida, você pode usar seu software de planilha para encontrar os desvios padrão da inclinação, a interceptação y e o valor y calculado. Para determinar os algarismos significativos para os valores da equação, seguimos uma regra simples. O último algarismo significativo de cada valor corresponde à primeira casa decimal significativa de seu desvio padrão.
Assim, arredondamos a inclinação para a casa dos milésimos e a interceptação y para a casa dos décimos, e arredondamos os desvios padrão para corresponder. Nossa inclinação é de 0,167 +/- 0,003 litros por Kelvin, e nossa interceptação em y é de -40,6 +/- 1,2 litros. Qualquer valor y calculado será arredondado para a décima casa e será +/- 0,8 litros. Esta equação descreve a relação entre a temperatura e o volume de um gás.
Neste laboratório, você criará um conjunto de dados de variáveis dependentes e independentes medindo o diâmetro e a circunferência de vários tamanhos de béqueres. Em seguida, você usará esses dados para criar um gráfico de dispersão e realizar uma regressão linear, tendo em mente a importância de algarismos significativos. Você também praticará habilidades de laboratório, como filtrar e medir o volume usando pipetas, prestando atenção à incerteza nas medições e análises.

medição científica e habilidades de laboratório

Gráficos bidimensionais
Alguns experimentos de química assumem a forma de alterar diretamente uma propriedade do sistema que você está estudando, também conhecida como variável independente, como temperatura, e medir os efeitos em outra propriedade, também conhecida como variável dependente, como volume. Uma vez que os dados tenham sido coletados, a interação entre os dois parâmetros deve ser quantificada - ou convertida em uma forma que possa ser avaliada - e comparada a outras relações.
Gráficos bidimensionais podem ser usados para derivar certos tipos de relações matemáticas entre duas propriedades ou para estabelecer que tal relação não existe entre elas. A análise determinará como a variável dependente muda em resposta à variável independente. No exemplo de ajuste da temperatura de um líquido ou gás e monitoramento de mudanças em seu volume, a temperatura é a variável independente e o volume é a variável dependente.
Para fazer um gráfico bidimensional, cada ponto de dados deve ter um valor conhecido como coordenada para as variáveis dependentes e independentes. A variável independente é plotada no eixo x e a variável dependente é plotada no eixo y. Esses gráficos são facilmente feitos em software de planilha, que também pode ser usado para analisar os dados plotados.
Ajuste de curva
Uma vez que um conjunto de dados tenha sido plotado em um gráfico bidimensional, o ajuste de curva pode ser usado para gerar uma equação, ou função, para a variável dependente em termos da variável independente. As funções representam um modelo matemático que melhor modela os dados dos quais é derivado. O ajuste de curva é a técnica de encontrar uma função que produz uma linha que é uma boa correspondência para o padrão de pontos de dados. O software de planilha possui várias ferramentas de ajuste de curva, o que é chamado de 'melhor ajuste'. Geralmente, essa é uma análise de regressão linear de mínimos quadrados, embora a maioria dos softwares também ofereça regressão não linear de mínimos quadrados.
A precisão da equação linear de melhor ajuste pode ser verificada substituindo os valores x para os pontos de dados e comparando os resultados 'teóricos' da equação com os valores y reais dos pontos de dados. O software de planilha normalmente calcula o valor do coeficiente de determinação (R2) para a função, que mostra o quão bem a função corresponde aos pontos de dados. Quanto mais próximo o valor de R2 estiver de 1, melhor será o ajuste para uma regressão linear. Outras funções têm métodos mais especializados para determinar o quão bem o ajuste da função é aos dados.
Determinar a incerteza dos valores dependentes calculados a partir da função de melhor ajuste exigiria técnicas complicadas de "propagação de erros". No entanto, é possível calcular a incerteza dentro da equação na forma do desvio padrão para a inclinação e a interceptação y de uma função de melhor ajuste. Isso geralmente é realizado com uma ferramenta diferente da usada para gerar um gráfico bidimensional.
desvio padrão
O desvio padrão descreve a quantidade de variação presente em um conjunto de valores. O desvio padrão da população (σ) é usado quando há dados de cada membro de uma população finita, como a massa de cada bolinha em um saco de bolinhas de gude. O desvio padrão da amostra é usado para todos os outros casos e é o cálculo do desvio padrão padrão no software de planilha. 1 Você pode supor que 'desvio padrão' se refere ao desvio padrão da amostra.
Presume-se que o erro de medição aleatório siga uma distribuição aproximadamente "normal", onde cerca de 68% de um conjunto de valores estão dentro de um intervalo de um desvio padrão em cada lado da média, 95% dos valores estão dentro de dois desvios padrão em cada lado da média e 99,7% dos valores estão dentro de três desvios padrão em cada lado da média. Assim, o desvio padrão é uma maneira útil de descrever erro e incerteza.
A equação para o desvio padrão da amostra é:
<img src="/CDNSource/lm/labs/42/42_Concepts_1.jpg">
Nesta equação, N é o número de valores; <img src="/CDNSource/lm/labs/42/30034eq02.jpg"> é a média (ou média) dos valores; e xi representa cada valor individual. Assim, para calcular o desvio padrão manualmente, calcule a média do conjunto de valores, subtraia a média de cada valor, eleve cada diferença ao quadrado, some as diferenças quadradas, divida a soma total por um a menos que o número de valores e tire a raiz quadrada do quociente. Quanto mais próximo s de zero, menor a variação entre os valores. Se os valores forem inseridos no software de planilha, o desvio padrão pode ser calculado de dentro do software.
O número de algarismos significativos em um desvio padrão depende de quais valores ele se destina. Ao comunicar o desvio-padrão para um grupo de pontos de dados obtidos nas mesmas condições, deve ser determinado primeiro o número adequado de algarismos significativos no valor médio. O desvio padrão é então arredondado para o mesmo número de casas decimais que a média. Para um conjunto de volumes com quatro algarismos significativos, uma média de 15,361 mL e um desvio padrão de 0,2313, a média e o desvio padrão seriam relatados como 15,36 mL ± 0,23 mL.
Ao relatar o desvio padrão da média e da interceptação y para uma função de melhor ajuste determinada pela análise de mínimos quadrados, que é o método usual para software de planilha, a primeira casa decimal do desvio padrão é o último algarismo significativo da média ou interceptação y. Assim, o desvio padrão deve ser arredondado para uma casa decimal significativa, e a inclinação ou interceptação y deve ser arredondada para a casa decimal correspondente. Por exemplo, se a inclinação for 0,1691 L·K-1 e tiver um desvio padrão de 0,00512, a inclinação deve ser relatada como 0,169 L·K-1 ± 0,005 L·K-1.
Se o desvio padrão da inclinação ou interceptação y for muito menor do que seu valor correspondente que seguir esta regra daria números mais significativos para a inclinação ou interceptação y do que os dados de medição originais permitiriam, então, em vez disso, determine os números significativos da inclinação ou interceptação y a partir dos valores x e y e arredonde o desvio padrão para um algarismo significativo. Assim, para uma inclinação de 0,1691 L·K-1, um desvio padrão de 0,0000512 e valores x e y com quatro algarismos significativos, a inclinação deve ser relatada como 0,1691 L·K-1 ± 0,00005 L·K-1. Nesse caso, não há problema em que o valor calculado e seu desvio padrão tenham um número diferente de casas decimais. Lembre-se de que o método dos mínimos quadrados usa os valores x e y para calcular a inclinação e a interceptação y.
Ao relatar o desvio padrão como um intervalo de incerteza, certifique-se de observar especificamente que a incerteza representa um desvio padrão. Isso diz ao leitor que há cerca de 68% de chance de que o valor real de uma medida esteja dentro desse intervalo da média, assumindo uma distribuição normal. Muitas vezes, pode ser mais adequado relatar a incerteza como dois desvios padrão da média, pois isso aumenta a probabilidade para cerca de 95%. Para fazer isso, basta multiplicar o desvio padrão por dois antes de arredondá-lo para o número apropriado de algarismos significativos.
Referências
<ol style="font-size:17px">
	<li style="font-size:17px">Harris, D.C. (2015). Análise Química Quantitativa. Nova York, NY: W.H. Freeman and Company.</li>
</ol>

Medição Científica e Habilidades de Laboratório

Gráficos bidimensionais
Alguns experimentos de química assumem a forma de alterar diretamente uma propriedade do sistema que você está estudando, também conhecida como ...

Education

JoVE Lab Manual

Química

graphs, curve fit, and standard deviation

Two-dimensional graphs help students compare an independent variable with a dependent variable in science data. In a chemistry example, temperature is the independent variable and volume is the dependent variable. After data are collected, the relationship between the two measurements must be quantified so it can be evaluated and compared with other relationships. A graph can also show when no relationship exists between the two variables.

To make a two-dimensional graph, each data point needs a coordinate for both variables. The independent variable is plotted on the x-axis, and the dependent variable is plotted on the y-axis. These plots are often made in spreadsheet software, which can also be used to analyze the data. This helps show how the dependent variable changes in response to the independent variable.

Once the data are plotted, curve fitting can turn the pattern into an equation, or function. A function is a mathematical model that best matches the data it comes from. Curve fitting looks for a line or curve that fits the data points well. In many cases, spreadsheet tools use a best-fit method based on linear least-squares regression, although nonlinear least-squares regression is also available in most software.

The quality of a best-fit line can be checked by comparing the x values in the equation with the actual y values from the data. Spreadsheet software usually reports the coefficient of determination, or R squared, which shows how well the function matches the plotted points. A value closer to 1 means a better fit for linear regression. Finding uncertainty for values calculated from the equation can be more complicated, but standard deviation can be used to describe the uncertainty in the slope and y-intercept of a best-fit line.

Standard deviation shows how much values vary in a set. Population standard deviation, written as sigma, is used when data are available for every member of a finite population, such as the mass of each marble in a bag. Sample standard deviation, written as s, is used in all other cases and is the default in spreadsheet software. For that reason, standard deviation usually means sample standard deviation. Random measurement error is assumed to follow a roughly normal distribution, where about 68% of values fall within one standard deviation of the mean, 95% fall within two, and 99.7% fall within three.

The sample standard deviation can also be calculated by hand. First find the mean. Then subtract the mean from each value, square each difference, add the squared differences, divide by one less than the number of values, and take the square root. The closer s is to zero, the less variation there is in the data. Spreadsheet software can also calculate standard deviation directly.

When reporting standard deviation, the number of significant figures depends on what it describes. For a group of measurements taken under the same conditions, the standard deviation is rounded to the same number of decimal places as the mean. For example, 15.361 mL with a standard deviation of 0.2313 should be reported as 15.36 mL ± 0.23 mL. When reporting the standard deviation of a slope or y-intercept from least-squares analysis, the first decimal place of the standard deviation becomes the last significant figure of the value. So 0.1691 L·K-1 with a standard deviation of 0.00512 is reported as 0.169 L·K-1 ± 0.005 L·K-1.

If the standard deviation is much smaller than the slope or y-intercept, this rule can give more significant figures than the original x and y data allow. In that case, use the significant figures from the data and round the standard deviation to one significant figure. For example, 0.1691 L·K-1 with a standard deviation of 0.0000512 and x and y values with four significant figures is reported as 0.1691 L·K-1 ± 0.00005 L·K-1. It is important to state whether the uncertainty represents one standard deviation. One standard deviation gives about a 68% chance that the true value lies within that range. Two standard deviations may be a better choice when a wider uncertainty range is needed, since that raises the chance to about 95%.

Graphs, Curve Fit, and Standard Deviation

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uv-vis spectroscopy of dyes

How Dyes Absorb Visible Light

Dyes absorb visible light because their molecular structure lets them interact with specific wavelengths. When light hits a substance, some light is absorbed and the rest is reflected or transmitted. The color we see depends on the wavelength that is reflected. A blue substance reflects blue light, around 430 to 480 nm, and absorbs the complementary orange region, around 590 to 630 nm. Compounds that do not absorb in the visible range appear colorless to the human eye. Light also has energy and...

identifying alcohols

Alcohol Tests for Primary, Secondary, and Tertiary

Alcohol tests can help sort alcohols by structure and reactivity. Alcohols are organic molecules with a hydroxyl group attached to an alkyl or aryl group, written as ROH. If the carbon with the hydroxyl group has one R group, the alcohol is primary. If it has two R groups, it is secondary. If it has three R groups, it is tertiary. Alcohols can also be aromatic when they contain a benzene ring. Phenol is the simplest aromatic alcohol. To identify an alcohol, chemists can use its chemical...

chemistry

Vídeos abrangentes e focados no currículo para instrutores e alunos de cursos introdutórios de laboratório de química

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Learn how to plot independent and dependent variables, use best-fit lines, and report standard deviation to describe uncertainty in science data.

Graphs, Curve Fit, and Standard Deviation | JoVE K12

Prepare-se para ensinar Medição Científica e Habilidades de Laboratório com este guia do instrutor de 3 minutos. Abrange as principais etapas de preparação para o laboratório Chemistry no JoVE Lab Manual.

Preparação do Instrutor para Medição Científica e Habilidades de Laboratório em Chemistry - 5.7K+ visualizações | JoVE Lab Manual

Siga os procedimentos passo a passo de Medição Científica e Habilidades de Laboratório neste vídeo de protocolo estudantil de 15 minutos. Parte da coleção Chemistry no JoVE Lab Manual.

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Scientific Laws and Theories

Scientific Laws

In science, a law is defined as a concise, verbal or mathematical, statement that summarizes a vast number of experimental observations. ...

The Scientific Method

Chemistry is an empirical science. Scientists often pose questions to understand the chemistry in everyday life and seek answers to these questions. To ...

CHEM.1.A

What is an Experiment?

An experiment is a planned activity carried out under controlled conditions. The purpose of an experiment is to investigate the relationship between two ...

Study Design in Statistics

A study design is a set of techniques that allow a researcher to collect and analyze data from different variables defined for a specific research ...

Observational Studies

Observational studies are a type of analytical study where researchers observe events without any interventions. In other words, the researcher does not ...

Experimental Designs

An experimental design is a systematic process that allows researchers to evaluate the relationship between dependent and independent variables. There are ...

Randomized Experiments

The randomization process involves assigning study participants randomly to experimental or control groups based on their probability of being equally ...

Crossover Experiments

Crossover experiments, also called the repeated-measurements design, is a study design in which all experimental units are exposed to all treatments in ...

Controls in Experiments

When conducting an experiment, it is crucial to have control to reduce bias and accurately measure the dependent variables. It also marks the results more ...

Bias

Bias refers to any tendency that prevents a question from being considered unprejudiced. In research, bias occurs when one outcome or answer is selected ...

Blinding

Blinding is a commonly used method of not telling participants which treatment a subject is receiving. Blinding is a critical part of a randomized control ...

CHEM.1.B

Proper Personal Protective Equipment

Robert M. Rioux &#38; William A Elliott, Pennsylvania State University, University Park, PA
Hazards are many and varied in the laboratory, but the right ...

Emergency Eyewash and Shower Stations

Robert M. Rioux, Pennsylvania State University, University Park, PA
The Occupational Safety and Health Administration (OSHA) mandates that an emergency ...

Electrical Safety

Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Among the many hazards present in the laboratory, electrical ...

Working with Centrifuges

Source: Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Earth's gravitational force is capable of separating ...

Working with Hot and Cold Sources

Source: Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Working with extreme temperatures, both high and low, ...

guidelines-in-case-of-a-laboratory-emergency

Guidelines in Case of a Laboratory Emergency

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
The most common laboratory emergencies include chemical spills, ...

chemical-storage-categories-hazards-and-compatibilities

Chemical Storage: Categories, Hazards And Compatibilities

Source: Robert M. Rioux &amp; Taslima A. Zaman, Pennsylvania State University, University Park, PA
While the use of various chemicals in experimental ...

Safe Handling of Mineral Acids

Source: Robert M. Rioux &#38; Taslima A. Zaman, Pennsylvania State University, University Park, PA
A mineral acid (or inorganic acid) is defined as a ...

Handling Chemical Spills

Source: Robert M. Rioux &#38; Taslima Zaman, Pennsylvania State University, University Park, PA
Since chemicals are commonly used for laboratory research ...

Proper Use of Autoclaves

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
Autoclaving is one of the most commonly used methods in the ...

Fume Hoods and Laminar Flow Cabinets

Robert M. Rioux &amp; William A. Elliott, Pennsylvania State University, University Park, PA
Fume hoods and laminar flow cabinets are engineering controls ...

handling-air-and-water-sensitive-chemicals-using-a-schlenk-line

Handling Air- and Water-Sensitive Chemicals Using a Schlenk Line

Robert M Rioux, Ajay Sathe, Zhifeng Chen, Pennsylvania State University, University Park, PA
The use of reagents sensitive to oxidation or moisture ...

proper-operation-of-vacuum-based-equipment

Proper Operation of Vacuum Based Equipment

Source: Robert M. Rioux, Ajay Sathe &#38; Zhifeng Chen, Pennsylvania State University, University Park, PA
Vacuum is required for a number of laboratory ...

Operating the Glovebox

Robert M Rioux, Ajay Sathe, Zhifeng Chen, Pennsylvania State University, University Park, PA 
A glovebox is an isolated enclosure that is designed to ...

operation-of-high-pressure-reactor-vessels

Operation of High-pressure Reactor Vessels

Robert M Rioux, Pennsylvania State University, University Park, PA
The use of gases in a synthetic chemistry laboratory is essential for carrying out a ...

CHEM.1.C

Lab Techniques

Measurement Accuracy
The ability to repeat an experiment and get the same results, or reproducibility, is essential in scientific research. However, ...

Two-Dimensional Graphs
Some chemistry experiments take the form of directly changing one property of the system that you are studying otherwise known as ...

Proper Lab Notebook Keeping

Organic Lab Notebook Keeping
Record keeping is a vital skill in science and organic chemistry labs. Without proper notes, descriptions, quantities, or ...

Basic Organic Chemistry Techniques

Glassware in Organic Chemistry
There are standard pieces of glassware that are used in the organic chemistry lab. Beakers and Erlenmeyer flasks are ...

Measurement: Standard Units

Every measurement provides three kinds of information: the size or magnitude of the measurement (a number), a standard of comparison for the measurement ...

Measurement: Derived Units

The International System of Units or SI system, by international agreement, has fixed measurement units for seven fundamental properties: length, mass, ...

uncertainty-in-measurement-accuracy-and-precision

Uncertainty in Measurement: Accuracy and Precision

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy ...

uncertainty-in-measurement-reading-instruments

Uncertainty in Measurement: Reading Instruments

Counting is the type of measurement that is free from uncertainty, provided the number of objects being counted does not change during the process. Such ...

CHEM.1.E

Scientific and engineering practices. The student, for at least 40% of instructional time, asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to answer questions, explain phenomena, or design solutions using appropriate tools and models. The student is expected to:

Graphs, Curve Fit, and Standard Deviation

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