Find the mean deviation about the mean for the data:
4, 7, 8, 9, 10, 12, 13, 17
3
Find the mean deviation about the mean for the data:
38, 70, 48, 40, 42, 55, 63, 46, 54, 44
8.4
Find the mean deviation about the median for the data:
13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17
2.33
Find the mean deviation about the median for the data:
36, 72, 46, 42, 60, 45, 53, 46, 51, 49
7
Find the mean deviation about the mean for the following data:
\(x_i\): 5, 10, 15, 20, 25
\(f_i\): 7, 4, 6, 3, 5
6.32
Find the mean deviation about the mean for the following data:
\(x_i\): 10, 30, 50, 70, 90
\(f_i\): 4, 24, 28, 16, 8
16
Find the mean deviation about the median for the following data:
\(x_i\): 5, 7, 9, 10, 12, 15
\(f_i\): 8, 6, 2, 2, 2, 6
3.23
Find the mean deviation about the median for the following data:
\(x_i\): 15, 21, 27, 30, 35
\(f_i\): 3, 5, 6, 7, 8
5.1
Find the mean deviation about the mean for the grouped data:
Income per day (₹): 0–100, 100–200, 200–300, 300–400, 400–500, 500–600, 600–700, 700–800
No. of persons: 4, 8, 9, 10, 7, 5, 4, 3
157.92
Find the mean deviation about the mean for the grouped data:
Height (cm): 95–105, 105–115, 115–125, 125–135, 135–145, 145–155
No. of boys: 9, 13, 26, 30, 12, 10
11.28
Find the mean deviation about the median for the following data:
Marks: 0–10, 10–20, 20–30, 30–40, 40–50, 50–60
No. of girls: 6, 8, 14, 16, 4, 2
10.34
Calculate the mean deviation about median age for the age distribution of 100 persons:
Age (years): 16–20, 21–25, 26–30, 31–35, 36–40, 41–45, 46–50, 51–55
No. of persons: 5, 6, 12, 14, 26, 12, 16, 9
7.35
Find the mean and variance for the data:
6, 7, 10, 12, 13, 4, 8, 12
Mean = 9, Variance = 9.25
Find the mean and variance for the first \(n\) natural numbers.
Mean = \(\dfrac{n+1}{2}\), Variance = \(\dfrac{n^{2}-1}{12}\)
Find the mean and variance for the first 10 multiples of 3.
Mean = 16.5, Variance = 74.25
Find the mean and variance for the following data:
\(x_i\): 6, 10, 14, 18, 24, 28, 30
\(f_i\): 2, 4, 7, 12, 8, 4, 3
Mean = 19, Variance = 43.4
Find the mean and variance for the following data:
\(x_i\): 92, 93, 97, 98, 102, 104, 109
\(f_i\): 3, 2, 3, 2, 6, 3, 3
Mean = 100, Variance = 29.09
Find the mean and standard deviation using the short-cut method:
\(x_i\): 60, 61, 62, 63, 64, 65, 66, 67, 68
\(f_i\): 2, 1, 12, 29, 25, 12, 10, 4, 5
Mean = 64, Standard deviation = 1.69
Find the mean and variance for the following frequency distribution:
Classes: 0–30, 30–60, 60–90, 90–120, 120–150, 150–180, 180–210
Frequencies: 2, 3, 5, 10, 3, 5, 2
Mean = 107, Variance = 2276
Find the mean and variance for the following frequency distribution:
Classes: 0–10, 10–20, 20–30, 30–40, 40–50
Frequencies: 5, 8, 15, 16, 6
Mean = 27, Variance = 132
Find the mean, variance and standard deviation using the short-cut method:
Height (cm): 70–75, 75–80, 80–85, 85–90, 90–95, 95–100, 100–105, 105–110, 110–115
No. of children: 3, 4, 7, 7, 15, 9, 6, 6, 3
Mean = 93, Variance = 105.58, Standard deviation = 10.27
The diameters of circles in a design are given below. Calculate the mean and standard deviation:
Diameters (mm): 33–36, 37–40, 41–44, 45–48, 49–52
No. of circles: 15, 17, 21, 22, 25
Mean = 5.55, Standard deviation = 43.5
The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
4, 8
The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12 and 14, find the remaining two observations.
6, 8
The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
24, 12
Given that \(\bar{x}\) is the mean and \(\sigma^{2}\) is the variance of \(n\) observations \(x_{1}, x_{2}, ..., x_{n}\). Prove that the mean and variance of the observations \(ax_{1}, ax_{2}, ax_{3}, ..., ax_{n}\) are \(a\bar{x}\) and \(a^{2}\sigma^{2}\), respectively, where \(a \neq 0\).
Mean = \(a\bar{x}\), Variance = \(a^{2}\sigma^{2}\)
The mean and standard deviation of 20 observations were found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
(i) If wrong item is omitted.
(ii) If it is replaced by 12.
(i) 10.1, 1.99
(ii) 10.2, 1.98
The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
20, 3.036