{
"query": {
"display": "desviación estándar $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178$$",
"symbolab_question": "STATISTICS#std 0.154,0.15,0.146,0.162,0.162,0.152,0.156,0.152,0.158,0.152,0.178"
},
"solution": {
"level": "PERFORMED",
"subject": "Statistics",
"topic": "std",
"subTopic": "Other",
"default": "0.00862…"
},
"steps": {
"type": "interim",
"title": "Desviación estándar de $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178:{\\quad}0.00862…$$",
"steps": [
{
"type": "definition",
"title": "Desviación estándar",
"text": "La desviación estándar, $$\\sigma\\left(X\\right)$$, es la raíz cuadrada de la varianza:$${\\quad}\\sigma\\left(X\\right)=\\sqrt{\\frac{\\sum_{i=1}^{n}\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}$$"
},
{
"type": "interim",
"title": "Calcular la varianza:$${\\quad}0.00007…$$",
"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, $$\\bar{x}:{\\quad}0.15654…$$",
"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:$${\\quad}\\sum_{i=1}^{n}a_{i}=1.722$$",
"steps": [
{
"type": "step",
"primary": "Tomar la suma de $$0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178$$",
"result": "0.154+0.15+0.146+0.162+0.162+0.152+0.156+0.152+0.158+0.152+0.178"
},
{
"type": "step",
"primary": "Simplificar",
"result": "1.722"
}
],
"meta": {
"interimType": "Take Sum Of Set Title 0Eq"
}
},
{
"type": "interim",
"title": "Calcular el número de elementos en el conjunto de datos:$${\\quad}n=11$$",
"input": "0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178",
"steps": [
{
"type": "step",
"primary": "Contar el número de elementos en el conjunto de datos",
"result": "\\begin{Bmatrix}0.154&0.15&0.146&0.162&0.162&0.152&0.156&0.152&0.158&0.152&0.178\\\\1&2&3&4&5&6&7&8&9&10&11\\end{Bmatrix}"
},
{
"type": "step",
"primary": "El número de elementos en el conjunto de datos es",
"result": "11"
}
],
"meta": {
"interimType": "Compute Number Terms Specific 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCD7Qw6ykTeoDaFv1B9imsi8iLOVs7thsurDEKJBF4k9Qo3kqf235XlFxYuGx8Gy2hsjEFqG8NOzmVbJlpWdfroRj7VOYytIpJPPs4EzRLbvJdbA+zX4bD3u3gx65o2NJhPYoOhfsW9ZWKHmZzdFhfiao77Bxg2amlnUTVH8UTTMulp2NUfRfMM4x2VWu3OA1PTdawJC3M/zbiG+X807f8Xw"
}
},
{
"type": "interim",
"title": "Dividir la suma entre el número de elementos y simplificar:$${\\quad}0.15654…$$",
"steps": [
{
"type": "step",
"primary": "Divide la suma por el número de términos:$${\\quad}\\frac{\\sum_{i=1}^{n}a_{i}}{n}=\\frac{1.722}{11}$$",
"result": "\\frac{1.722}{11}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "0.15654…"
}
],
"meta": {
"interimType": "Compute The Average Title 0Eq"
}
},
{
"type": "step",
"result": "=0.15654…"
}
],
"meta": {
"interimType": "Arithmetic Mean Top 1Eq"
}
},
{
"type": "interim",
"title": "Calcular $$\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2:{\\quad}0.00074…$$",
"steps": [
{
"type": "step",
"primary": "Tomar la suma de $$\\left(0.154-0.15654…\\right)^{2},\\:\\left(0.15-0.15654…\\right)^{2},\\:\\left(0.146-0.15654…\\right)^{2},\\:\\left(0.162-0.15654…\\right)^{2},\\:\\left(0.162-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.156-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.158-0.15654…\\right)^{2},\\:\\left(0.152-0.15654…\\right)^{2},\\:\\left(0.178-0.15654…\\right)^{2}$$",
"result": "\\left(0.154-0.15654…\\right)^{2}+\\left(0.15-0.15654…\\right)^{2}+\\left(0.146-0.15654…\\right)^{2}+\\left(0.162-0.15654…\\right)^{2}+\\left(0.162-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.156-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.158-0.15654…\\right)^{2}+\\left(0.152-0.15654…\\right)^{2}+\\left(0.178-0.15654…\\right)^{2}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "0.00074…"
}
],
"meta": {
"interimType": "Generic Compute Title 1Eq"
}
},
{
"type": "interim",
"title": "Calcular el número de elementos en el conjunto de datos:$${\\quad}n=11$$",
"input": "0.154,\\:0.15,\\:0.146,\\:0.162,\\:0.162,\\:0.152,\\:0.156,\\:0.152,\\:0.158,\\:0.152,\\:0.178",
"steps": [
{
"type": "step",
"primary": "Contar el número de elementos en el conjunto de datos",
"result": "\\begin{Bmatrix}0.154&0.15&0.146&0.162&0.162&0.152&0.156&0.152&0.158&0.152&0.178\\\\1&2&3&4&5&6&7&8&9&10&11\\end{Bmatrix}"
},
{
"type": "step",
"primary": "El número de elementos en el conjunto de datos es",
"result": "11"
}
],
"meta": {
"interimType": "Compute Number Terms Specific 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3K5O+fXcZudMI+14ZU5ZwQp9wy31HdDJRMD4uiaWz07swGDq7iA6052JsaffwjYCD7Qw6ykTeoDaFv1B9imsi8iLOVs7thsurDEKJBF4k9Qo3kqf235XlFxYuGx8Gy2hsjEFqG8NOzmVbJlpWdfroRj7VOYytIpJPPs4EzRLbvJdbA+zX4bD3u3gx65o2NJhPYoOhfsW9ZWKHmZzdFhfiao77Bxg2amlnUTVH8UTTMulp2NUfRfMM4x2VWu3OA1PTdawJC3M/zbiG+X807f8Xw"
}
},
{
"type": "interim",
"title": "Calcular $$Var\\left(X\\right)=\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}:{\\quad}0.00007…$$",
"steps": [
{
"type": "step",
"primary": "$$\\frac{\\sum_{i=1}^n\\left(x_i-\\bar{x}\\right)^2}{n-1}=\\frac{0.00074…}{10}$$",
"result": "\\frac{0.00074…}{10}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "0.00007…"
}
],
"meta": {
"interimType": "Compute The Variance Title 0Eq"
}
},
{
"type": "step",
"result": "0.00007…"
}
],
"meta": {
"interimType": "Variance Top 1Eq"
}
},
{
"type": "interim",
"title": "Calcular $$\\sigma\\left(X\\right)=\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}:{\\quad}0.00862…$$",
"steps": [
{
"type": "step",
"primary": "La varianza es $$0.00007…$$ , por lo tanto $$\\sqrt{\\sum_{i=1}^{n}\\frac{\\left(x_{i}-\\bar{x}\\right)^2}{n-1}}=\\sqrt{0.00007…}$$",
"result": "\\sqrt{0.00007…}"
},
{
"type": "step",
"primary": "Simplificar",
"result": "0.00862…"
}
],
"meta": {
"interimType": "Compute The STDV Title 0Eq"
}
},
{
"type": "step",
"result": "0.00862…"
}
]
}
}
Solución
desviación estándar
Solución
Pasos de solución
Calcular la varianza:
Calcular