The height of five children are as follows: $$ 61 \mathrm{~cm}, 70 \mathrm{~cm}, 54 \mathrm{~cm}, 81 \mathrm{~cm} $$, and 42 cm Find the mean $$ (M) $$ and standard deviation (SD)

Question

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(61+70+54+81+42\right)}{5}$$
- step1: Remove the parentheses:
  $$\frac{61+70+54+81+42}{5}$$
- step2: Add the numbers:
  $$\frac{308}{5}$$
Calculate or simplify the expression $$ \sqrt(((61 - M)^2 + (70 - M)^2 + (54 - M)^2 + (81 - M)^2 + (42 - M)^2) / 5) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\sqrt{\frac{\left(\left(61-M\right)^{2}+\left(70-M\right)^{2}+\left(54-M\right)^{2}+\left(81-M\right)^{2}+\left(42-M\right)^{2}\right)}{5}}$$
- step1: Remove the parentheses:
  $$\sqrt{\frac{\left(61-M\right)^{2}+\left(70-M\right)^{2}+\left(54-M\right)^{2}+\left(81-M\right)^{2}+\left(42-M\right)^{2}}{5}}$$
- step2: Add the terms:
  $$\sqrt{\frac{19862-616M+5M^{2}}{5}}$$
- step3: Use the properties of radicals:
  $$\frac{\sqrt{19862-616M+5M^{2}}}{\sqrt{5}}$$
- step4: Simplify:
  $$\frac{\sqrt{99310-3080M+25M^{2}}}{5}$$
To find the mean $$ M $$ and standard deviation (SD) of the heights of the five children, we will follow these steps:

### Step 1: Calculate the Mean $$ M $$

The mean is calculated using the formula:

$$
M = \frac{\text{Sum of all heights}}{\text{Number of heights}}
$$

Given the heights: $$ 61 \, \text{cm}, 70 \, \text{cm}, 54 \, \text{cm}, 81 \, \text{cm}, 42 \, \text{cm} $$

Calculating the sum:

$$
\text{Sum} = 61 + 70 + 54 + 81 + 42 = 308
$$

Now, we divide by the number of children (5):

$$
M = \frac{308}{5} = 61.6 \, \text{cm}
$$

### Step 2: Calculate the Standard Deviation (SD)

The standard deviation is calculated using the formula:

$$
SD = \sqrt{\frac{\sum (x_i - M)^2}{N}}
$$

Where:
- $$ x_i $$ are the individual heights,
- $$ M $$ is the mean,
- $$ N $$ is the number of heights.

Substituting the values:

$$
SD = \sqrt{\frac{(61 - 61.6)^2 + (70 - 61.6)^2 + (54 - 61.6)^2 + (81 - 61.6)^2 + (42 - 61.6)^2}{5}}
$$

Calculating each term:

1. $$ (61 - 61.6)^2 = (-0.6)^2 = 0.36 $$
2. $$ (70 - 61.6)^2 = (8.4)^2 = 70.56 $$
3. $$ (54 - 61.6)^2 = (-7.6)^2 = 57.76 $$
4. $$ (81 - 61.6)^2 = (19.4)^2 = 376.36 $$
5. $$ (42 - 61.6)^2 = (-19.6)^2 = 384.16 $$

Now, summing these values:

$$
\sum (x_i - M)^2 = 0.36 + 70.56 + 57.76 + 376.36 + 384.16 = 889.2
$$

Now, substituting back into the SD formula:

$$
SD = \sqrt{\frac{889.2}{5}} = \sqrt{177.84} \approx 13.33
$$

### Final Results

- Mean $$ M \approx 61.6 \, \text{cm} $$
- Standard Deviation $$ SD \approx 13.33 \, \text{cm} $$
The mean height is approximately 61.6 cm, and the standard deviation is approximately 13.33 cm.

The height of five children are as follows: \( 61 \mathrm{~cm}, 70 \mathrm{~cm}, 54 \mathrm{~cm}, 81 \mathrm{~cm} \), and 42 cm Find the mean \( (M) \) and standard deviation (SD)

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Statistics Questions