Male and female populations of humpback whales under 80 years old are shown below by age:

Age: 0-9; Male: 12; Female: 5
age: 10-16; Male: 16; female: 7
Age: 20-29; Male: 12; Female: 17
Age: 30-39; Male: 19; females 18
age: 40-49; Male: 25; female: 24
Age: 50-59; male: 20; female: 22
Age: 60-69; male: 17; female: 16
Age: 70-79; male: 14; female: 13

Approximate the population mean and standard deviation for both male and female.

To approximate the population mean and standard deviation for both the male and female humpback whale populations under 80 years old, we'll follow these steps:

### Steps to Calculate Population Mean and Standard Deviation:
1. **Calculate the Midpoint of Each Age Group**: For each age group, find the midpoint, which will be used as a representative value for that group.
2. **Calculate the Population Mean**: Use the midpoints and the number of whales in each group to calculate the mean for both males and females.
3. **Calculate the Population Standard Deviation**: Use the midpoints, the number of whales in each group, and the mean to calculate the standard deviation.

### Data:
- **Age Groups and Midpoints**: 
  - 0-9: Midpoint = 4.5
  - 10-19: Midpoint = 13
  - 20-29: Midpoint = 24.5
  - 30-39: Midpoint = 34.5
  - 40-49: Midpoint = 44.5
  - 50-59: Midpoint = 54.5
  - 60-69: Midpoint = 64.5
  - 70-79: Midpoint = 74.5

- **Male Population**: 12, 16, 12, 19, 25, 20, 17, 14
- **Female Population**: 5, 7, 17, 18, 24, 22, 16, 13

Now, let's calculate the mean and standard deviation for both male and female populations.

### 1. Calculate the Population Mean:
The population mean is given by:

\[
\mu = \frac{\sum{(x_i \cdot f_i)}}{\sum{f_i}}
\]

Where \(x_i\) is the midpoint of the age group, and \(f_i\) is the frequency of the population in that group.

### 2. Calculate the Population Standard Deviation:
The population standard deviation is given by:

\[
\sigma = \sqrt{\frac{\sum{f_i \cdot (x_i - \mu)^2}}{\sum{f_i}}}
\]

Let me calculate these for both males and females.The approximate population mean and standard deviation for humpback whales under 80 years old are as follows:

- **Males**:
  - Mean Age: 41.14 years
  - Standard Deviation: 21.37 years

- **Females**:
  - Mean Age: 44.41 years
  - Standard Deviation: 18.96 years

These values indicate that on average, the female population is slightly older, with a smaller spread (less variability) in age compared to the male population.

Would you like more details on the calculations or have any further questions?

Here are five related questions:
1. How does the age distribution of males compare to that of females?
2. What could be the ecological implications of these age distributions?
3. How does the standard deviation affect the population study?
4. Could the data indicate anything about the longevity of male vs. female humpback whales?
5. How would including whales over 80 years old affect these calculations?

**Tip:** Standard deviation helps you understand how spread out the ages are from the mean, which can reveal population stability or variability.

Explore how to approximate the population mean and standard deviation for male and female humpback whales under 80 years old. Detailed calculations are provided, showing the average age and variability in age distribution. Learn more about these statistics in ecological studies.

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