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{
  "topic": "percentage-error",
  "input_types": [
    "float",
    "float"
  ],
  "input_labels": [
    "X",
    "Y"
  ],
  "input_descriptions": [
    "True Value",
    "Observed Value"
  ],
  "input_units": [
    "%",
    "%"
  ],
  "input_values": [
    "",
    ""
  ],
  "type": "Calculate",
  "title": "Percentage Error",
  "category": "Percentage",
  "formula_mathjax": "\\( \\dfrac { X - Y }{ X } \\times 100 \\)",
  "formula": "(( X - Y )/ X )*100",
  "op_label": "Percent Error",
  "output_unit": "%",
  "explanation": "Online Percentage Error calculator helps you determine the percentage error between an experimental value and a theoretical value. Enter the experimental and theoretical values in the input fields, and the answer is calculated as a percentage error.",
  "content": "<h2>Calculating Percentage Error</h2>\n<p>Percentage error is a measure used to express the difference between a true or theoretical value and an observed or experimental value. It is often used in scientific experiments, statistical analysis, and quality control to assess the accuracy of measurements or predictions.</p>\n<p>The formula to calculate the percentage error is expressed as:</p>\n<p>\\( \\text{Percentage Error} = \\dfrac { X - Y }{ X } \\times 100 \\)</p>\n<p>where:</p>\n<ul>\n<li><b>X</b> represents the true or theoretical value.</li>\n<li><b>Y</b> represents the observed or experimental value.</li>\n</ul><p>By subtracting the observed value from the true value, dividing by the true value, and then multiplying by 100, you obtain the percentage error.</p>\n\n<h2>Examples</h2><p>The following examples demonstrate how to calculate the percentage error between a true value and an observed value using the given formula.</p><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">1.</span> A chemist measures the boiling point of a liquid to be 102°C, but the true boiling point is known to be 100°C. What is the percentage error in the chemist's measurement?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li><b>X</b> = 100°C (True boiling point)</li><li><b>Y</b> = 102°C (Observed boiling point)</li></ul><p>The formula to calculate the percentage error is:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { X - Y }{ X } \\times 100 \\)</p><p>Substituting the given values into the formula:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 100 - 102 }{ 100 } \\times 100 \\)</p><p>Simplifying further:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { -2 }{ 100 } \\times 100 = -2\\% \\)</p><p class=\"tabspace answer\">Therefore, the percentage error in the chemist's measurement is -2%, indicating that the observed value is 2% higher than the true value.</p></div><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">2.</span> A student calculates the density of a metal to be 7.8 g/cm³, but the true density of the metal is 8.0 g/cm³. What is the percentage error in the student's calculation?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li><b>X</b> = 8.0 g/cm³ (True density)</li><li><b>Y</b> = 7.8 g/cm³ (Observed density)</li></ul><p>The formula to calculate the percentage error is:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { X - Y }{ X } \\times 100 \\)</p><p>Substituting the given values into the formula:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 8.0 - 7.8 }{ 8.0 } \\times 100 \\)</p><p>Simplifying further:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 0.2 }{ 8.0 } \\times 100 = 2.5\\% \\)</p><p class=\"tabspace answer\">Therefore, the percentage error in the student's calculation is 2.5%, indicating that the observed value is 2.5% lower than the true value.</p></div><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">3.</span> In a physics experiment, the acceleration due to gravity is measured to be 9.6 m/s², while the standard value is 9.8 m/s². What is the percentage error in the measured value?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li><b>X</b> = 9.8 m/s² (True value of acceleration due to gravity)</li><li><b>Y</b> = 9.6 m/s² (Observed value)</li></ul><p>The formula to calculate the percentage error is:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { X - Y }{ X } \\times 100 \\)</p><p>Substituting the given values into the formula:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 9.8 - 9.6 }{ 9.8 } \\times 100 \\)</p><p>Simplifying further:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 0.2 }{ 9.8 } \\times 100 = 2.04\\% \\)</p><p class=\"tabspace answer\">Therefore, the percentage error in the measured value of gravity is 2.04%, indicating that the observed value is 2.04% lower than the true value.</p></div><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">4.</span> A laboratory scale shows a mass of 250 grams for a sample, but the true mass of the sample is 245 grams. What is the percentage error in the mass measurement?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li><b>X</b> = 245 grams (True mass)</li><li><b>Y</b> = 250 grams (Observed mass)</li></ul><p>The formula to calculate the percentage error is:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { X - Y }{ X } \\times 100 \\)</p><p>Substituting the given values into the formula:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { 245 - 250 }{ 245 } \\times 100 \\)</p><p>Simplifying further:</p><p class=\"tabspace\">\\( \\text{Percentage Error} = \\dfrac { -5 }{ 245 } \\times 100 = -2.04\\% \\)</p><p class=\"tabspace answer\">Therefore, the percentage error in the mass measurement is -2.04%, indicating that the observed value is 2.04% higher than the true value.</p></div>"
}