{
"query": {
"display": "varianza $$82,\\:44,\\:67,\\:52,\\:120$$",
"symbolab_question": "STATISTICS#variance 82,44,67,52,120"
},
"solution": {
"level": "PERFORMED",
"subject": "Statistics",
"topic": "variance",
"subTopic": "Other",
"default": "902"
},
"steps": {
"type": "interim",
"title": "Varianza muestral de $$82,\\:44,\\:67,\\:52,\\:120:{\\quad}902$$",
"steps": [
{
"type": "definition",
"title": "Varianza muestral",
"text": "La varianza muestral mide cuán disperos están los datos en la muestra.<br/>para un conjunto de datos $$x_{1},\\:\\ldots\\:,\\:x_{n}$$ (n elementos) con un promedio $$\\bar{x}$$, $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}$$"
},
{
"type": "interim",
"title": "Calcular el promedio, $$\\bar{x}:{\\quad}73$$",
"steps": [
{
"type": "definition",
"title": "Media aritmética",
"text": "La media aritmética (promedio) es la suma de los valores en un conjunto divididos entre el número de elementos del mismo<br/>Si nuestro conjunto de datos contiene los valores $$a_{1},\\:\\ldots\\:,\\:a_{n}$$ (n elementos) entonces el promedio $$=\\frac{1}{n}\\sum_{i=1}^{n}a_{i}\\:$$"
},
{
"type": "interim",
"title": "Calcular la suma del conjunto de datos:$${\\quad}\\sum_{i=1}^{n}a_{i}=365$$",
"steps": [
{
"type": "step",
"primary": "Tomar la suma de $$82,\\:44,\\:67,\\:52,\\:120$$",
"result": "82+44+67+52+120"
},
{
"type": "step",
"primary": "Simplificar",
"result": "365"
}
],
"meta": {
"interimType": "Take Sum Of Set Title 0Eq"
}
},
{
"type": "interim",
"title": "Calcular el número de elementos en el conjunto de datos:$${\\quad}n=5$$",
"input": "82,\\:44,\\:67,\\:52,\\:120",
"steps": [
{
"type": "step",
"primary": "Contar el número de elementos en el conjunto de datos",
"result": "\\begin{Bmatrix}82&44&67&52&120\\\\1&2&3&4&5\\end{Bmatrix}"
},
{
"type": "step",
"primary": "El número de elementos en el conjunto de datos es",
"result": "5"
}
],
"meta": {
"interimType": "Compute Number Terms Specific 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAPH6cV8dmqzpETS6G76c46bLMWDFrHTOo2D5n/oMNXp3ql8XXPq6bNQlMm+36iNhnLY+n9yRYL18twGmFmLUgF0QPk693F7T1rHO/o1DvzgLlQ3ukLy2xkO/4xxxVu82fiRwJI5cO7eavcUUt4Nto1"
}
},
{
"type": "interim",
"title": "Dividir la suma entre el número de elementos y simplificar:$${\\quad}73$$",
"steps": [
{
"type": "step",
"primary": "Divide la suma por el número de términos:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{365}{5}$$",
"result": "\\frac{365}{5}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "73"
}
],
"meta": {
"interimType": "Compute The Average Title 0Eq"
}
},
{
"type": "step",
"result": "=73"
}
],
"meta": {
"interimType": "Arithmetic Mean Top 1Eq"
}
},
{
"type": "interim",
"title": "Calcular $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}3608$$",
"steps": [
{
"type": "step",
"primary": "Tomar la suma de $$\\left(82-73\\right)^{2},\\:\\left(44-73\\right)^{2},\\:\\left(67-73\\right)^{2},\\:\\left(52-73\\right)^{2},\\:\\left(120-73\\right)^{2}$$",
"result": "\\left(82-73\\right)^{2}+\\left(44-73\\right)^{2}+\\left(67-73\\right)^{2}+\\left(52-73\\right)^{2}+\\left(120-73\\right)^{2}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "3608"
}
],
"meta": {
"interimType": "Generic Compute Title 1Eq"
}
},
{
"type": "interim",
"title": "Calcular el número de elementos en el conjunto de datos:$${\\quad}n=5$$",
"input": "82,\\:44,\\:67,\\:52,\\:120",
"steps": [
{
"type": "step",
"primary": "Contar el número de elementos en el conjunto de datos",
"result": "\\begin{Bmatrix}82&44&67&52&120\\\\1&2&3&4&5\\end{Bmatrix}"
},
{
"type": "step",
"primary": "El número de elementos en el conjunto de datos es",
"result": "5"
}
],
"meta": {
"interimType": "Compute Number Terms Specific 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCAPH6cV8dmqzpETS6G76c46bLMWDFrHTOo2D5n/oMNXp3ql8XXPq6bNQlMm+36iNhnLY+n9yRYL18twGmFmLUgF0QPk693F7T1rHO/o1DvzgLlQ3ukLy2xkO/4xxxVu82fiRwJI5cO7eavcUUt4Nto1"
}
},
{
"type": "interim",
"title": "Calcular $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}902$$",
"steps": [
{
"type": "step",
"primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{3608}{4}$$",
"result": "\\frac{3608}{4}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "902"
}
],
"meta": {
"interimType": "Compute The Variance Title 0Eq"
}
},
{
"type": "step",
"result": "902"
}
]
}
}
Solución
varianza
Solución
Pasos de solución
Calcular el promedio,
Calcular
Calcular el número de elementos en el conjunto de datos:
Calcular