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

Traducir los hallazgos experimentales de puntos de datos a representaciones visuales, como gráficos, es esencial para determinar la relación entre dos o más propiedades. Estas propiedades se denominan variables. Cuando hay dos variables, un gráfico creado a partir de los datos se denomina bidimensional. El gráfico tiene dos ejes. La variable independiente se representa en el eje x y la variable dependiente se representa en el eje y.
Tomemos, por ejemplo, estos datos de muestra para la temperatura y el volumen de un gas. El volumen del gas depende de la temperatura. Por lo tanto, trazaríamos la temperatura medida en el eje x y el volumen en el eje y.
Cuando hay varios puntos de datos con el mismo valor x, como si medimos el volumen varias veces a una temperatura, también calculamos la desviación estándar de esas mediciones. La desviación estándar es un valor estadístico, que indica la cantidad de variación presente en un conjunto de valores.
La desviación estándar se calcula utilizando esta fórmula, donde n es el número de puntos de datos, x bar es el valor medio de los puntos de datos y xi representa cada punto de datos individual. Puede calcular la desviación estándar a mano, o un programa de hoja de cálculo puede calcularla automáticamente. Cuanto más cerca esté la desviación estándar de 0, más cerca estarán los puntos de datos del valor medio. Si la desviación estándar es igual a 0, todos los puntos de datos introducidos tienen el mismo valor.
Echemos un vistazo a los valores medios y las desviaciones estándar de las mediciones de volumen a cada temperatura en nuestro conjunto de datos. Podemos resumir cada conjunto de puntos de datos como la media, más o menos la desviación estándar. Determinamos las cifras significativas para cada media a partir de las mediciones correspondientes y redondeamos los valores medios en consecuencia.
La desviación estándar de cada grupo debe tener el mismo número de decimales que la media, por lo que redondeamos cada desviación estándar a las centésimas. Para determinar gráficamente la relación entre dos variables, podemos ajustar los datos con una función de mejor ajuste.
La función es generada automáticamente por el software de la hoja de cálculo y puede tomar la forma de una línea de tendencia lineal, una función polinómica o una función exponencial o logarítmica. En el caso de nuestros datos de temperatura y volumen, la relación es lineal. Por lo tanto, los puntos de datos se ajustan mediante una regresión de mínimos cuadrados lineales. Su programa de hoja de cálculo devolverá la ecuación para la línea de mejor ajuste y un valor de r cuadrado. Cuanto más cerca esté el valor de r cuadrado de 1, mejor será el ajuste de los datos.
A continuación, puede utilizar el software de su hoja de cálculo para encontrar las desviaciones estándar de la pendiente, la intersección con el eje y y y el valor y calculado. Para determinar las cifras significativas de los valores de la ecuación, seguimos una regla simple. La última cifra significativa de cada valor corresponde al primer decimal significativo de su desviación estándar.
Por lo tanto, redondeamos la pendiente hasta el lugar de las milésimas y la intersección con el eje y hasta el lugar de las décimas, y redondeamos las desviaciones estándar para que coincidan. Nuestra pendiente es de 0,167 +/- 0,003 litros por Kelvin, y nuestra intersección con el eje y es de -40,6 +/- 1,2 litros. Cualquier valor y calculado se redondeará a las décimas y será de +/- 0,8 litros. Esta ecuación describe la relación entre la temperatura y el volumen de un gas.
En este laboratorio, creará un conjunto de datos de variables dependientes e independientes midiendo el diámetro y la circunferencia de varios tamaños de vasos de precipitados. A continuación, utilizará estos datos para crear un diagrama de dispersión y realizar una regresión lineal, teniendo en cuenta la importancia de las cifras significativas. También practicarás habilidades de laboratorio, como filtrar y medir el volumen con pipetas, prestando atención a la incertidumbre en las mediciones y el análisis.

medición científica y habilidades de laboratorio

Gráficos bidimensionales
Algunos experimentos de química toman la forma de cambiar directamente una propiedad del sistema que se está estudiando, también conocida como la variable independiente, como la temperatura, y medir los efectos sobre otra propiedad, también conocida como la variable dependiente, como el volumen. Una vez que se han recopilado los datos, la interacción entre los dos parámetros debe cuantificarse, o convertirse en una forma que pueda evaluarse, y compararse con otras relaciones.
Los gráficos bidimensionales se pueden utilizar para derivar ciertos tipos de relaciones matemáticas entre dos propiedades o para establecer que no existe tal relación entre ellas. En última instancia, el análisis determinará cómo cambia la variable dependiente en respuesta a la variable independiente. En el ejemplo de ajuste de la temperatura de un líquido o gas y monitoreo de cambios en su volumen, la temperatura es la variable independiente y el volumen es la variable dependiente.
Para hacer un gráfico bidimensional, cada punto de datos debe tener un valor conocido como coordenada para las variables dependientes e independientes. La variable independiente se representa en el eje x y la variable dependiente se representa en el eje y. Estos gráficos se realizan fácilmente en un software de hoja de cálculo, que también se puede utilizar para analizar los datos trazados.
Ajuste de curvas
Una vez que se ha trazado un conjunto de datos en un gráfico bidimensional, el ajuste de curva se puede utilizar para generar una ecuación, o función, para la variable dependiente en términos de la variable independiente. Las funciones representan un modelo matemático que modela mejor los datos de los que se derivan. El ajuste de curva es la técnica de encontrar una función que produce una línea que coincide bien con el patrón de puntos de datos. El software de hojas de cálculo tiene varias herramientas de ajuste de curvas, lo que se conoce como "mejor ajuste". Por lo general, se trata de un análisis de regresión de mínimos cuadrados lineales, aunque la mayoría del software también ofrece una regresión de mínimos cuadrados no lineales.
La precisión de la ecuación lineal de mejor ajuste se puede comprobar introduciendo los valores x de los puntos de datos y comparando los resultados "teóricos" de la ecuación con los valores y reales de los puntos de datos. El software de hoja de cálculo normalmente calculará el valor del coeficiente de determinación (R2) para la función, que muestra qué tan bien coincide la función con los puntos de datos. Cuanto más se acerque el valor de R2 a 1, mejor será el ajuste para una regresión lineal. Otras funciones tienen métodos más especializados para determinar qué tan bien se ajusta la función a los datos.
La determinación de la incertidumbre de los valores dependientes calculados a partir de la función de mejor ajuste requeriría técnicas complicadas de "propagación de errores". Sin embargo, es posible calcular la incertidumbre dentro de la ecuación en forma de la desviación estándar tanto para la pendiente como para la intersección con el eje y de una función de mejor ajuste. Por lo general, esto se realiza con una herramienta diferente a la que se usa para generar un gráfico bidimensional.
desviación estándar
La desviación estándar describe la cantidad de variación presente en un conjunto de valores. La desviación estándar de la población (σ) se utiliza cuando hay datos de cada miembro de una población finita, como la masa de cada canica en una bolsa de canicas. La desviación estándar de la muestra (s) se utiliza para todos los demás casos y es el cálculo de la desviación estándar predeterminada en el software de hoja de cálculo. 1 Se puede suponer que la "desviación estándar" se refiere a la desviación estándar de la muestra.
Se supone que el error de medición aleatorio sigue una distribución más o menos "normal", en la que aproximadamente el 68% de un conjunto de valores se encuentra dentro de un rango de una desviación estándar a cada lado de la media, el 95% de los valores se encuentran dentro de dos desviaciones estándar a cada lado de la media, y el 99,7% de los valores se encuentran dentro de tres desviaciones estándar a cada lado de la media. Por lo tanto, la desviación estándar es una forma útil de describir el error y la incertidumbre.
La ecuación para la desviación estándar de la muestra es:
<img src="/CDNSource/lm/labs/42/42_Concepts_1.jpg">
En esta ecuación, N es el número de valores; <img src="/CDNSource/lm/labs/42/30034eq02.jpg"> es el promedio (o media) de los valores; y xi representa cada valor individual. Por lo tanto, para calcular la desviación estándar a mano, calcule la media del conjunto de valores, reste la media de cada valor, eleve cada diferencia al cuadrado, sume las diferencias al cuadrado, divida la suma total por uno menos que el número de valores y tome la raíz cuadrada del cociente. Cuanto más cerca esté s de cero, menor será la variación entre los valores. Si los valores se introducen en el software de la hoja de cálculo, la desviación estándar se puede calcular desde el software.
El número de cifras significativas en una desviación estándar depende de los valores para los que se trate. Al informar de la desviación estándar de un grupo de puntos de datos tomados en las mismas condiciones, primero se debe determinar el número apropiado de cifras significativas en el valor medio. A continuación, la desviación estándar se redondea al mismo número de decimales que la media. Para un conjunto de volúmenes con cuatro cifras significativas, una media de 15,361 mL y una desviación estándar de 0,2313, la media y la desviación estándar se informarían como 15,36 mL ± 0,23 mL.
Al informar de la desviación estándar de la media y la intersección con el eje y para una función de mejor ajuste determinada por el análisis de mínimos cuadrados, que es el método habitual para el software de hojas de cálculo, el primer decimal de la desviación estándar es la última cifra significativa de la media o la intersección con el eje y. Por lo tanto, la desviación estándar debe redondearse a un decimal significativo, y la pendiente o intersección con el eje y debe redondearse al decimal correspondiente. Por ejemplo, si la pendiente es 0,1691 L·K-1 y tiene una desviación estándar de 0,00512, la pendiente debe notificarse como 0,169 L·K-1 ± 0,005 L·K-1.
Si la desviación estándar de la pendiente o de la intersección con el eje y es mucho menor que su valor correspondiente que seguir esta regla daría cifras más significativas a la pendiente o a la intersección con el eje y de lo que permitirían los datos de medición originales, entonces determine las cifras significativas de la pendiente o la intersección con el eje y a partir de los valores x e y y redondee la desviación estándar a una cifra significativa. Por lo tanto, para una pendiente de 0,1691 L·K-1, una desviación estándar de 0,0000512 y valores de x e y con cuatro cifras significativas, la pendiente debe informarse como 0,1691 L·K-1 ± 0,00005 L·K-1. En este caso, está bien que el valor calculado y su desviación estándar tengan un número diferente de decimales. Ten en cuenta que el método de mínimos cuadrados utiliza los valores x e y para calcular la pendiente y la intersección con el eje y.
Al informar la desviación estándar como un rango de incertidumbre, asegúrese de tener en cuenta específicamente que la incertidumbre representa una desviación estándar. Esto le dice al lector que hay aproximadamente un 68% de posibilidades de que el valor real de una medición caiga dentro de ese rango de la media, suponiendo una distribución normal. A menudo puede ser más adecuado informar de la incertidumbre como dos desviaciones estándar de la media, ya que eso aumenta la probabilidad a aproximadamente el 95%. Para ello, basta con multiplicar la desviación estándar por dos antes de redondearla al número adecuado de cifras significativas.
Referencias
<ol style="font-size:17px">
	<li style="font-size:17px">Harris, D.C. (2015). Análisis químico cuantitativo. Nueva York, NY: W.H. Freeman and Company.</li>
</ol>

Medición Científica y Habilidades de Laboratorio

Gráficos bidimensionales
Algunos experimentos de química toman la forma de cambiar directamente una propiedad del sistema que está estudiando, también conocida como ...

Education

Educación

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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scientific measurement and lab skills

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Gas Laws Behind the Ideal Gas Equation

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acid and base concentrations

Acids, Bases, and pH in Titration

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buffers

Buffer pH and the Common Ion Effect

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solubility

Solubility in Water and Other Solvents

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Flame Test Metal Color Clues

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beer's law

Absorbance Spectra and Beer-Lambert Analysis

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concentration dependence

Reaction Order and Rate Constants

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Reaction Rate, Arrhenius, and Catalysts

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proper lab notebook keeping

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basic organic chemistry techniques

Organic Lab Glassware and Safe Handling

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melting points

How Impurities Change Melting Points

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boiling points

Boiling Point and Vapor Pressure

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recrystallization

Purifying Crystals by Solubility

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extraction

Separating Mixtures with Solvent Extraction

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simple distillation

Separating Liquids by Boiling Point

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steam distillation

Steam Distillation and Low-Boiling Separation

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TLC Separation by Polarity

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Separating Mixtures by Polarity

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identification-of-unknown-aldehydes-and-ketones

Carbonyl Tests for Aldehydes and Ketones

Aldehydes and ketones are carbonyl compounds, which means they contain a C=O functional group. A ketone has two alkyl or aryl groups attached to the carbonyl carbon. Acetone is the simplest ketone, with two methyl groups. An aldehyde is similar, but it has at least one hydrogen attached to the carbonyl carbon. Formaldehyde is the simplest aldehyde, and acetaldehyde has one hydrogen and one methyl group. The carbonyl carbon in both aldehydes and ketones is electrophilic because the oxygen atom...

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

Videos completos y enfocados en el plan de estudios, tanto para los instructores como para los estudiantes de los cursos de laboratorio de introducción a la 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

Prepárese para enseñar Medición Científica y Habilidades de Laboratorio con esta guía para instructores de 3 minutos. Cubre los pasos clave de preparación para el laboratorio Chemistry en JoVE Lab Manual.

Preparación del instructor para Medición Científica y Habilidades de Laboratorio en Chemistry - Más de 5.7K visualizaciones | JoVE Lab Manual

Siga los procedimientos paso a paso de Medición Científica y Habilidades de Laboratorio en este video de protocolo estudiantil de 15 minutos. Parte de la colección Chemistry en JoVE Lab Manual.

Protocolo estudiantil de Medición Científica y Habilidades de Laboratorio en Chemistry - Más de 20K visualizaciones | JoVE Lab Manual

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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