Let's define each term with examples and explanations.

1. Mean

The mean, or average, is the sum of all data points divided by the number of data points.

Example: Consider the data set: $5, 8, 12, 15, 20$.

Calculation: $\text{Mean} = \frac{5 + 8 + 12 + 15 + 20}{5} = \frac{60}{5} = 12$

2. Median

The median is the middle value in a sorted data set. If the number of data points is even, the median is the average of the two middle values.

Example: For the data set $5, 8, 12, 15, 20$, which is already sorted, the median is $12$, as it's the middle number.

If the data set was $5, 8, 12, 15$, the median would be: $\frac{8 + 12}{2} = 10$

3. Mode

The mode is the most frequently occurring value in a data set.

Example: For the data set $5, 8, 8, 12, 15$, the mode is $8$ since it appears more often than the other numbers.

4. Variance

Variance measures how spread out the data points are from the mean. It is calculated by finding the average of the squared differences between each data point and the mean.

Example: For the data set $5, 8, 12, 15, 20$, with a mean of $12$:

1. Calculate each deviation from the mean:
   $(5-12)^2 = 49$, $(8-12)^2 = 16$, $(12-12)^2 = 0$, $(15-12)^2 = 9$, $(20-12)^2 = 64$.

2. Find the average of these squared deviations: $\text{Variance} = \frac{49 + 16 + 0 + 9 + 64}{5} = \frac{138}{5} = 27.6$

5. Standard Deviation

Standard deviation is the square root of the variance, providing a measure of the average distance of each data point from the mean.

Example: The variance we calculated was $27.6$.

The standard deviation is: $\text{Standard Deviation} = \sqrt{27.6} \approx 5.26$

Summary:

• Mean: Average value.
• Median: Middle value when data is sorted.
• Mode: Most frequent value.
• Variance: Average of squared deviations from the mean.
• Standard Deviation: Square root of variance, showing spread from the mean.

Would you like more details or have any questions?

5 Related Questions:

1. How do mean and median differ in skewed distributions?
2. What are the practical applications of standard deviation in statistics?
3. How do outliers affect the variance and standard deviation?
4. Can a data set have more than one mode? Explain with examples.
5. Why is variance calculated using squared differences?

Tip:

For normally distributed data, approximately 68% of the values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This is known as the "68-95-99.7 rule."