Arithmetic Mean Problems

1The numbers 4.47 and 10.15 are added to a set of 5 numbers whose mean is 7.31. What is the mean of the new set of numbers?

2A dentist records the number of cavities in 100 children from a school. The information obtained is summarized in the following table:

No. of cavities fi ni
0 25 0.25
1 20 0.2
2 x z
3 15 0.15
4 y 0.05

1. Complete the table to obtain the values of x, y, z.

2. Calculate the average number of cavities.

3Complete the missing data in the following statistical table:

xi fi Fi ni
1 4   0.08
2 4    
3   16 0.16
4 7   0.14
5 5 28  
6   38  
7 7 45  
8      

Also, calculate the mean.

4Consider the following data: 3, 8, 4, 10, 6, 2.

1. Calculate its mean and variance.

2. If all the above data was multiplied by 3, what would the new mean and variance be?

5 The result of throwing two dice 120 times is represented by the table:

Sums 2 3 4 5 6 7 8 9 10 11 12
No. of Times 3 8 9 11 20 19 16 13 11 6 4

1. Calculate the mean and standard deviation.

2. Find the percentage of values in the interval (x − σ, x + σ).

6The heights of the players (in centimeters) from a basketball team are represented by the table:

Height [170, 175) [175, 180) [180, 185) [185, 190) [190, 195) [195, 2.00)
No. of players 1 3 4 8 5 2

Calculate:

1. The mean.

2. The median.

3. The standard deviation.

4. How many players are above the mean plus one standard deviation?

7The result of throwing one dice 200 times is represented by the following table:

  1 2 3 4 5 6
fi a 32 35 33 b 35

Determine the value of a and b knowing that the average score is 3.6.

8Given the absolute cumulative frequency table:

Age Fi
[0, 2) 4
[2, 4) 11
[4, 6) 24
[6, 8) 34
[8, 10) 40

1. Calculate the arithmetic mean and standard deviation.

2. Calculate the difference between the values that are the 10 central ages?


1

The numbers 4.47 and 10.15 are added to a set of 5 numbers whose mean is 7.31. What is the mean of the new set of numbers?

Arithmetic Mean Exercise


2

A dentist records the number of cavities in 100 children from a school. The information obtained is summarized in the following table:

No. of cavities fi ni
0 25 0.25
1 20 0.2
2 x z
3 15 0.15
4 y 0.05

1. Complete the table to obtain the values of x, y, z.

The sum of the relative frequencies must be equal to 1:

0.25 + 0.2 + z + 0.15 + 0.05 = 1

0.65 + z = 1 z = 0.35

The relative frequency of the data equals the absolute frequency divided by 100, which is the sum of the absolute frequencies.

Arithmetic Mean Exercise

Arithmetic Mean Exercise

No. of caries fi ni fi · ni
0 25 0.25 0
1 20 0.2 20
2 35 0.35 70
3 15 0.15 45
4 5 0.05 20
      155

2. Calculate the average number of cavities.

Arithmetic Mean Solution


3

Complete the missing data in the following statistical table:

xi fi Fi ni
1 4   0.08
2 4    
3   16 0.16
4 7   0.14
5 5 28  
6   38  
7 7 45  
8      

Also, calculate the mean.


Table

First row:

F1 = 4Arithmetic Mean Operations

Second row:

F2 = 4 + 4 = 8      Arithmetic Mean Operations

Third row:

Arithmetic Mean Operations

Fourth row:

N4 = 16 + 7 = 23

Fifth row:

Arithmetic Mean Operations

Sixth row:

28 + n8 = 38      n8 = 10 Arithmetic Mean Operations

Seventh row:

Arithmetic Mean Operations

Eighth row:

N8 = N = 50 n8 = 50 − 45 = 5 Arithmetic Mean Operations


xi fi Fi ni xi · fi
1 4 4 0.08 4
2 4 8 0.08 8
3 8 16 0.16 24
4 7 23 0.14 28
5 5 28 0.1 25
6 10 38 0.2 60
7 7 45 0.14 49
8 5 50 0.1 40
  50     238

Arithmetic Mean

Arithmetic Mean Solution


4

Consider the following data: 3, 8, 4, 10, 6, 2.

1. Calculate its mean and variance.

2. If all the above data was multiplied by 3, what would the new mean and variance be?


xi xi2
2 4
3 9
4 16
6 36
8 64
10 100
33 229

1

Arithmetic Mean Calculations

2

Arithmetic Mean Operations


5

The result of throwing two dice 120 times is represented by the table:

Sums 2 3 4 5 6 7 8 9 10 11 12
No. of Times 3 8 9 11 20 19 16 13 11 6 4

1. Calculate the mean and standard deviation.

2. Find the percentage of values in the interval (x − σ, x + σ).


xi fi xi · fi xi2 · fi
2 3 6 12
3 8 24 72
4 9 36 144
5 11 55 275
6 20 120 720
7 19 133 931
8 16 128 1024
9 13 117 1053
10 11 110 1100
11 6 66 726
12 4 48 576
  120 843 6633

1

Arithmetic Mean Calculations

2

x − σ = 4.591 x + σ = 9.459

The values in the interval (4.591, 9.459) are those relating to the amounts of 5, 6, 7, 8 and 9.

11 + 20 + 19 + 16 + 13 = 79

Arithmetic Mean Operations


6

The heights of the players (in centimeters) from a basketball team are represented by the table:

Height [170, 175) [175, 180) [180, 185) [185, 190) [190, 195) [195, 2.00)
No. of players 1 3 4 8 5 2

Calculate:

1. The mean.

2. The median.

3. The standard deviation.

4. How many players are above the mean plus one standard deviation?


  xi fi Fi xi · fi xi2 · fi
[1.70, 1.75) 1.725 1 1 1.725 2.976
[1.75, 1.80) 1.775 3 4 5.325 9.453
[1.80, 1.85) 1.825 4 8 7.3 13.324
[1.85, 1.90) 1.875 8 16 15 28.128
[1.90, 1.95) 1.925 5 21 9.625 18.53
[1.95, 2.00) 1.975 2 23 3.95 7.802
    23   42.925 80.213

Mean

Arithmetic Mean Operations

Median

Median Calculations

Standard Deviation

Standard Deviation

4

x + σ = 1.866+ 0.077 = 1.943

This value belongs to a percentile that is in the penultimate interval.

Arithmetic Mean Problems

Arithmetic Mean Solution

There are only 3 players above x + σ.


7

The result of throwing one dice 200 times is represented by the following table:

  1 2 3 4 5 6
fi a 32 35 33 b 35

Determine the value of a and b knowing that the average score is 3.6.


xi fi xi · fi
1 a a
2 32 64
3 35 125
4 33 132
5 b 5b
6 35 210
  135 + a + b 511 + a + 5b

Arithmetic Mean Operations

Arithmetic Mean

a = 29 b = 36


8

Given the absolute cumulative frequency table:

Edad Fi
[0, 2) 4
[2, 4) 11
[4, 6) 24
[6, 8) 34
[8, 10) 40

1. Calculate the arithmetic mean and standard deviation.

2. Calculate the difference between the values that are the 10 central ages?


  xi fi Fi xi · fi xi2 · fi
[0, 2) 1 4 4 4 4
[2, 4) 3 7 11 21 63
[4, 6) 5 13 24 65 325
[6, 8) 7 10 34 70 490
[8, 10) 9 6 40 54 486
    40   214 1368

Mean and Standard Deviation

Arithmetic Mean

Arithmetic Mean and Deviation

2

Arithmetic Mean Operations

The 10 students represent 25% of the central distribution.

Central Distribution Diagram

Find: P37.5 y P62.5.

Statistical Operations

Statistical Operations

The 10 central ages are in the interval: [4.61, 6.2] .




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