To solve the problem, we will follow these steps:
### Step 1: Calculate the Mean (x̄)
To find the mean, we sum all the observations and divide by the number of observations.
Given data:
34, 66, 30, 38, 44, 50, 40, 60, 42, 51
**Calculation:**
\[
\text{Mean} (x̄) = \frac{34 + 66 + 30 + 38 + 44 + 50 + 40 + 60 + 42 + 51}{10}
\]
\[
= \frac{ 34 + 66 + 30 + 38 + 44 + 50 + 40 + 60 + 42 + 51 }{10} = \frac{ 405 }{10} = 45.5
\]
### Step 2: Calculate the Mean Deviation (MD)
The mean deviation is calculated using the formula:
\[
MD = \frac{\sum |x_i - x̄|}{n}
\]
where \( x_i \) are the observations, \( x̄ \) is the mean, and \( n \) is the number of observations.
**Calculating the absolute deviations:**
- For each observation, calculate \( |x_i - x̄| \):
- For 34: \( |34 - 45.5| = 11.5 \)
- For 66: \( |66 - 45.5| = 20.5 \)
- For 30: \( |30 - 45.5| = 15.5 \)
- For 38: \( |38 - 45.5| = 7.5 \)
- For 44: \( |44 - 45.5| = 1.5 \)
- For 50: \( |50 - 45.5| = 4.5 \)
- For 40: \( |40 - 45.5| = 5.5 \)
- For 60: \( |60 - 45.5| = 14.5 \)
- For 42: \( |42 - 45.5| = 3.5 \)
- For 51: \( |51 - 45.5| = 5.5 \)
**Sum of absolute deviations:**
\[
\sum |x_i - x̄| = 11.5 + 20.5 + 15.5 + 7.5 + 1.5 + 4.5 + 5.5 + 14.5 + 3.5 + 5.5 = 90
\]
**Mean Deviation Calculation:**
\[
MD = \frac{90}{10} = 9
\]
### Step 3: Calculate the Range
Now, we need to find the range between \( x̄ - MD \) and \( x̄ + MD \).
**Calculation:**
\[
x̄ - MD = 45.5 - 9 = 36.5
\]
\[
x̄ + MD = 45.5 + 9 = 54.5
\]
### Step 4: Count the Observations in the Range
Now we need to count how many observations lie between 36.5 and 54.5.
**Observations:**
- 34 (not in range)
- 66 (not in range)
- 30 (not in range)
- 38 (in range)
- 44 (in range)
- 50 (in range)
- 40 (in range)
- 60 (not in range)
- 42 (in range)
- 51 (in range)
**Counting:**
The observations that lie between 36.5 and 54.5 are: 38, 44, 50, 40, 42, and 51.
Total observations in range: **6**
### Final Answer
The number of observations lying between \( x̄ - MD \) and \( x̄ + MD \) is **6**.
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