Measures of Central Tendency and Dispersion - Class X

When all observations in a dataset are identical, the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) are all equal to that constant value.

A frequency distribution with two modes is called bimodal. Unimodal has one mode, and trimodal has three.

The geometric mean involves taking the nth root of the product of n numbers. If any number is negative, the root may be imaginary; if zero, the product is zero. Thus, it is typically defined for positive observations.

The geometric mean is not suitable for further algebraic treatment (like calculating the mean of means) because it is based on products rather than sums.

The median is a positional average and is not affected by extreme values (outliers) in the dataset, unlike the arithmetic mean.

When the mode is not clearly identifiable in a frequency distribution, the grouping method is used to determine the modal class or value.

Like the geometric mean, the harmonic mean is not suitable for further algebraic treatment because it is based on the reciprocal of the values.

Harmonic mean gives more weight to smaller values because of the reciprocal in its formula. A small number becomes large when reciprocated, dominating the calculation.

mean $=\dfrac{\sum {x _1}}{n}=\dfrac{90}{11}\approx 8.18$

If a constant is subtracted from every observation in a set, the mean of the set is also decreased by that same constant.

The mode is the value that appears most frequently. In the set 10, 19, 22, 16, 15, 18, 20, 18, 14, 18, 23, the number 18 appears three times, which is more than any other number.

The second quartile (Q2) is the median. First, sort the data: 10, 14, 15, 16, 18, 18, 18, 19, 20, 22, 23. With 11 observations, the median is the 6th value, which is 18.

The median (Md) divides the data into two halves (50%), the second quartile (Q2) is the median, the 5th decile (D5) is the median, and the 50th percentile (P50) is the median. Thus, Md = Q2 = D5 = P50.

Deciles divide data into 10 parts, so D3 represents 30% of the data. Percentiles divide data into 100 parts, so P30 also represents 30% of the data. Therefore, D3 = P30.

Given AM = (a+b)/2 = 6.5, so a+b = 13. Given GM = sqrt(ab) = 6, so ab = 36. The numbers 4 and 9 satisfy a+b = 13 and ab = 36.

The arithmetic mean is the sum of observations divided by the count. (10 + (-10)) / 2 = 0 / 2 = 0.

The sum of the first n natural numbers is n(n+1)/2. The average is the sum divided by n, which results in (n+1)/2.

By definition, the mode of a frequency distribution is the value that occurs most frequently, meaning it has the maximum frequency.

For a series with N observations, if N is odd, the median is the ((N+1)/2)th observation. Substituting N = 2n+1 gives ((2n+1+1)/2)th = (2n+2)/2 = (n+1)th observation.

Total runs in 4 innings = 4 * 32 = 128. Total runs needed for 5 innings = 5 * 50 = 250. Runs needed in 5th inning = 250 - 128 = 122.

In a moderately skewed distribution, the empirical relationship is: Mean - Mode = 3(Mean - Median). Given Mean = 10 and Mode = 7, we have 10 - 7 = 3(10 - Median), so 3 = 3(10 - Median), which means 1 = 10 - Median, so Median = 9.

Given, the mean annual salary of all employees in a company is $Rs\ 25,000$ & mean salaries of male and female employees are $Rs\ 27,000$ and $Rs\ 17,000$ respectively.

Quartile deviation divides the series into four equal parts and measures the distance average between the third and the first quartile. The first quartile is denoted as Q₁ and the third quartile is denoted as Q₃ . 

Q₁= 46 and Q₃=54

Quartile deviation = (Q₃-Q₁) /2

                             = (54-46)/2

                             = 8/2

                             = 4