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Summary
1.Measures of Central Tendency
Measures of central tendency are statistical measures that describe the center or typical value of a dataset. The most common measures are the mean, median, and mode.
Sample Mean
The sample mean is the average of a sample, calculated by summing all the values and dividing by the number of values.
Example: For the sample data [2, 4, 6, 8, 10], the sample mean is (2+4+6+8+10)/5 = 6.
Population Mean
The population mean is the average of an entire population, calculated similarly to the sample mean but using the entire population data.
Example: For the population data [3, 5, 7, 9, 11], the population mean is (3+5+7+9+11)/5 = 7.
2.Mean for Grouped Data (Discrete Value Data)
The mean for grouped data with discrete values is calculated by multiplying each value by its frequency, summing these products, and dividing by the total number of values.
Example: For the data values [1, 2, 3] with frequencies [2, 3, 1], the mean is (1*2 + 2*3 + 3*1) / (2+3+1) = 11/6 ≈ 1.83.
3.Mean for Grouped Data (Continuous Data)
The mean for grouped continuous data is calculated using the midpoints of the class intervals, multiplied by their frequencies, summed, and divided by the total number of values.
Example: For class intervals [0-10, 10-20] with frequencies [3, 2], the midpoints are [5, 15]. The mean is (5*3 + 15*2) / (3+2) = 45/5 = 9.
Median
The median is the middle value of a dataset when it is ordered. If the dataset has an even number of values, the median is the average of the two middle values.
Example: For the data [1, 3, 3, 6, 7, 8, 9], the median is 6.
Mode
The mode is the value that appears most frequently in a dataset.
Example: For the data [1, 2, 2, 3, 4], the mode is 2.
4.Measures of Dispersion
Measures of dispersion describe the spread or variability of a dataset. Common measures include range, variance, standard deviation, and interquartile range.
4.1Range
The range is the difference between the maximum and minimum values in a dataset.
Example: For the data [2, 4, 6, 8, 10], the range is 10 - 2 = 8.
4.2Variance
Variance measures the average squared deviation of each value from the mean.
Example: For the data [2, 4, 6, 8, 10], the variance is [(2-6)^2 + (4-6)^2 + (6-6)^2 + (8-6)^2 + (10-6)^2] / 5 = 8.
4.3Standard Deviation
Standard deviation is the square root of the variance, representing the average distance of each value from the mean.
Example: For the data [2, 4, 6, 8, 10], the standard deviation is √8 ≈ 2.83.
4.4Interquartile Range (IQR)
The IQR is the difference between the third quartile (Q3) and the first quartile (Q1), representing the middle 50% of the data.
Example: For the data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Q1 is 2.75 and Q3 is 7.25, so IQR = 7.25 - 2.75 = 4.5.
4.5First Quartile (Q1)
The first quartile is the value below which 25% of the data falls.
Example: For the data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Q1 is 2.75.
4.6Second Quartile (Q2)
The second quartile is the median, the value below which 50% of the data falls.
Example: For the data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Q2 is 5.5.
4.7Third Quartile (Q3)
The third quartile is the value below which 75% of the data falls.
Example: For the data [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Q3 is 7.25.
4.8Outlier
An outlier is a value that is significantly higher or lower than most of the values in a dataset.
Example: For the data [1, 2, 3, 4, 5, 100], the value 100 is an outlier.
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