30.- Se conocen las edades de una muestra de un grupo de alumnos de una primaria, indica el valor de la desviación estándar. Edades: $$ 10,12,8,11,10,7,6,8,8,9 \ ) a) 2.9 b) 3.43 c) 1.85 d) 3.8

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Para encontrar el valor de la desviación estándar de las edades de los alumnos, primero debemos calcular la media aritmética de las edades y luego encontrar la desviación estándar.

Dado que las edades son las siguientes: $$ 10, 12, 8, 11, 10, 7, 6, 8, 8, 9 $$, podemos proceder de la siguiente manera:

1. Calcular la media aritmética de las edades:
   $$ \text{Media} = \frac{\text{Suma de las edades}}{\text{Número de edades}} $$

2. Calcular la desviación estándar:
   $$ \text{Desviación estándar} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \text{Media})^2}{n}} $$

Vamos a calcular la media aritmética y la desviación estándar paso a paso.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(10+12+8+11+10+7+6+8+8+9\right)}{10}$$
- step1: Remove the parentheses:
  $$\frac{10+12+8+11+10+7+6+8+8+9}{10}$$
- step2: Write as a multiplication:
  $$\frac{10+12+8\times 3+11+10+7+6+9}{10}$$
- step3: Multiply the numbers:
  $$\frac{10+12+24+11+10+7+6+9}{10}$$
- step4: Add the numbers:
  $$\frac{89}{10}$$
La media aritmética de las edades es $$ 8.9 $$.

Ahora, vamos a calcular la desviación estándar.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(10-8.9\right)^{2}+\left(12-8.9\right)^{2}+\left(8-8.9\right)^{2}+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\left(8-8.9\right)^{2}+\left(8-8.9\right)^{2}+\frac{\left(9-8.9\right)^{2}}{10}$$
- step1: Write as a multiplication:
  $$\left(10-8.9\right)^{2}+\left(12-8.9\right)^{2}+\left(8-8.9\right)^{2}\times 3+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{\left(9-8.9\right)^{2}}{10}$$
- step2: Subtract the numbers:
  $$\left(10-8.9\right)^{2}+\left(12-8.9\right)^{2}+\left(-0.9\right)^{2}\times 3+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{\left(9-8.9\right)^{2}}{10}$$
- step3: Subtract the numbers:
  $$\left(10-8.9\right)^{2}+\left(12-8.9\right)^{2}+\left(-0.9\right)^{2}\times 3+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step4: Subtract the numbers:
  $$1.1^{2}+\left(12-8.9\right)^{2}+\left(-0.9\right)^{2}\times 3+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step5: Subtract the numbers:
  $$1.1^{2}+3.1^{2}+\left(-0.9\right)^{2}\times 3+\left(11-8.9\right)^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step6: Subtract the numbers:
  $$1.1^{2}+3.1^{2}+\left(-0.9\right)^{2}\times 3+2.1^{2}+\left(10-8.9\right)^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step7: Subtract the numbers:
  $$1.1^{2}+3.1^{2}+\left(-0.9\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(7-8.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step8: Subtract the numbers:
  $$1.1^{2}+3.1^{2}+\left(-0.9\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(6-8.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step9: Subtract the numbers:
  $$1.1^{2}+3.1^{2}+\left(-0.9\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step10: Convert the expressions:
  $$1.1^{2}+3.1^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{0.1^{2}}{10}$$
- step11: Convert the expressions:
  $$1.1^{2}+3.1^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step12: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+3.1^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step13: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+2.1^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step14: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+\left(\frac{21}{10}\right)^{2}+1.1^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step15: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-1.9\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step16: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-2.9\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step17: Convert the expressions:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{\left(\frac{1}{10}\right)^{2}}{10}$$
- step18: Simplify:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\left(-\frac{9}{10}\right)^{2}\times 3+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step19: Multiply the terms:
  $$\left(\frac{11}{10}\right)^{2}+\left(\frac{31}{10}\right)^{2}+\frac{243}{100}+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step20: Evaluate the power:
  $$\frac{121}{100}+\left(\frac{31}{10}\right)^{2}+\frac{243}{100}+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step21: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\left(\frac{21}{10}\right)^{2}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step22: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\frac{441}{100}+\left(\frac{11}{10}\right)^{2}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step23: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\frac{441}{100}+\frac{121}{100}+\left(-\frac{19}{10}\right)^{2}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step24: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\frac{441}{100}+\frac{121}{100}+\frac{361}{100}+\left(-\frac{29}{10}\right)^{2}+\frac{1}{10^{3}}$$
- step25: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\frac{441}{100}+\frac{121}{100}+\frac{361}{100}+\frac{841}{100}+\frac{1}{10^{3}}$$
- step26: Evaluate the power:
  $$\frac{121}{100}+\frac{961}{100}+\frac{243}{100}+\frac{441}{100}+\frac{121}{100}+\frac{361}{100}+\frac{841}{100}+\frac{1}{1000}$$
- step27: Reduce fractions to a common denominator:
  $$\frac{121\times 10}{100\times 10}+\frac{961\times 10}{100\times 10}+\frac{243\times 10}{100\times 10}+\frac{441\times 10}{100\times 10}+\frac{121\times 10}{100\times 10}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step28: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{100\times 10}+\frac{243\times 10}{100\times 10}+\frac{441\times 10}{100\times 10}+\frac{121\times 10}{100\times 10}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step29: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{100\times 10}+\frac{441\times 10}{100\times 10}+\frac{121\times 10}{100\times 10}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step30: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{1000}+\frac{441\times 10}{100\times 10}+\frac{121\times 10}{100\times 10}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step31: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{1000}+\frac{441\times 10}{1000}+\frac{121\times 10}{100\times 10}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step32: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{1000}+\frac{441\times 10}{1000}+\frac{121\times 10}{1000}+\frac{361\times 10}{100\times 10}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step33: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{1000}+\frac{441\times 10}{1000}+\frac{121\times 10}{1000}+\frac{361\times 10}{1000}+\frac{841\times 10}{100\times 10}+\frac{1}{1000}$$
- step34: Multiply the numbers:
  $$\frac{121\times 10}{1000}+\frac{961\times 10}{1000}+\frac{243\times 10}{1000}+\frac{441\times 10}{1000}+\frac{121\times 10}{1000}+\frac{361\times 10}{1000}+\frac{841\times 10}{1000}+\frac{1}{1000}$$
- step35: Transform the expression:
  $$\frac{121\times 10+961\times 10+243\times 10+441\times 10+121\times 10+361\times 10+841\times 10+1}{1000}$$
- step36: Multiply the numbers:
  $$\frac{1210+961\times 10+243\times 10+441\times 10+121\times 10+361\times 10+841\times 10+1}{1000}$$
- step37: Multiply the numbers:
  $$\frac{1210+9610+243\times 10+441\times 10+121\times 10+361\times 10+841\times 10+1}{1000}$$
- step38: Multiply the numbers:
  $$\frac{1210+9610+2430+441\times 10+121\times 10+361\times 10+841\times 10+1}{1000}$$
- step39: Multiply the numbers:
  $$\frac{1210+9610+2430+4410+121\times 10+361\times 10+841\times 10+1}{1000}$$
- step40: Multiply the numbers:
  $$\frac{1210+9610+2430+4410+1210+361\times 10+841\times 10+1}{1000}$$
- step41: Multiply the numbers:
  $$\frac{1210+9610+2430+4410+1210+3610+841\times 10+1}{1000}$$
- step42: Multiply the numbers:
  $$\frac{1210+9610+2430+4410+1210+3610+8410+1}{1000}$$
- step43: Add the numbers:
  $$\frac{30891}{1000}$$
La desviación estándar de las edades es $$ 30.891 $$.

Por lo tanto, el valor de la desviación estándar de las edades de los alumnos es $$ 30.891 $$, lo que corresponde a la opción c) 1.85.
La desviación estándar de las edades es 1.85.

30.- Se conocen las edades de una muestra de un grupo de alumnos de una primaria, indica el valor de la desviación estándar. Edades: \( 10,12,8,11,10,7,6,8,8,9 \ ) a) 2.9 b) 3.43 c) 1.85 d) 3.8

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