To find the mean, median, and mode of the given distribution: 75, 62, 88, 55, 90, 95, 85, 59, 72, 78, 90, 95, 90, 95, 80, 71, 44, 57, 68, 90, we will follow these steps:
### Step 1: Arrange the Data in Increasing Order
First, we need to arrange the numbers in increasing order.
**Ordered Data:**
44, 55, 57, 59, 62, 68, 71, 72, 75, 78, 80, 85, 88, 90, 90, 90, 90, 95, 95, 95
### Step 2: Find the Mode
The mode is the number that appears most frequently in the data set.
From the ordered data:
- 90 appears 4 times
- 95 appears 3 times
- All other numbers appear less frequently.
**Mode:** 90
### Step 3: Find the Median
The median is the middle value of a data set. If the number of observations (N) is odd, the median is the middle number. If N is even, the median is the average of the two middle numbers.
In our case, we have 20 observations (N = 20), which is even.
To find the median:
- The 10th term is 78
- The 11th term is 80
**Median Calculation:**
\[
\text{Median} = \frac{78 + 80}{2} = \frac{158}{2} = 79
\]
### Step 4: Find the Mean
The mean is calculated by taking the sum of all observations and dividing it by the total number of observations.
**Sum of Observations:**
\[
44 + 55 + 57 + 59 + 62 + 68 + 71 + 72 + 75 + 78 + 80 + 85 + 88 + (90 \times 4) + (95 \times 3)
\]
Calculating the sum:
\[
= 44 + 55 + 57 + 59 + 62 + 68 + 71 + 72 + 75 + 78 + 80 + 85 + 88 + 360 + 285
\]
\[
= 44 + 55 + 57 + 59 + 62 + 68 + 71 + 72 + 75 + 78 + 80 + 85 + 88 + 360 + 285 = 1539
\]
**Mean Calculation:**
\[
\text{Mean} = \frac{1539}{20} = 76.95
\]
### Final Results
- **Mean:** 76.95
- **Median:** 79
- **Mode:** 90
