The sum of the deviations of the variates 6, 8, 10, 16, 20, 24

Options

A. ` -1`
B. 1
C. 0
D. 4

Answer

To find the sum of the deviations of the variates 6, 8, 10, 16, 20, and 24, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) of the variates is calculated by summing all the values and dividing by the number of values.

\[ \text{Mean} (x̄) = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Calculating the sum of the variates:
\[ 6 + 8 + 10 + 16 + 20 + 24 = 84 \]

Now, count the number of observations:
\[ \text{Number of observations} = 6 \]

Now, calculate the mean:
\[ x̄ = \frac{84}{6} = 14 \]

### Step 2: Calculate the Deviations
The deviation for each variate is calculated as:
\[ \text{Deviation} = x - x̄ \]

Calculating the deviations for each variate:
- For 6: \( 6 - 14 = -8 \)
- For 8: \( 8 - 14 = -6 \)
- For 10: \( 10 - 14 = -4 \)
- For 16: \( 16 - 14 = 2 \)
- For 20: \( 20 - 14 = 6 \)
- For 24: \( 24 - 14 = 10 \)

### Step 3: List the Deviations
The deviations are:
- -8
- -6
- -4
- 2
- 6
- 10

### Step 4: Calculate the Sum of Deviations
Now, we will sum all the deviations:
\[ \text{Sum of deviations} = (-8) + (-6) + (-4) + 2 + 6 + 10 \]

Calculating step by step:
\[ (-8) + (-6) = -14 \]
\[ -14 + (-4) = -18 \]
\[ -18 + 2 = -16 \]
\[ -16 + 6 = -10 \]
\[ -10 + 10 = 0 \]

### Final Result
The sum of the deviations is:
\[ \text{Sum of deviations} = 0 \]