Notes on Arithmetic Mean | Grade 10 > Compulsory Mathematics > Statistics | KULLABS.COM

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The word "statistics " refers two meanings. In the singular sense, it deals with the collection, presentation, analysis and interpretation of numerical data and helps in making a decision. In the plural sense, it refers to the numerical facts and figures are sometimes known as statistical data.

### Measures of central tendency

A typical value which represents the characteristics of the entire mass of huge data is called the central value of the whole distribution. A measure of the central tendency is also known as a measure of location or an average.
The various measure of central tendency are as follow:

1. Arithmetic Mean (average)
2. Median
3. Mode

### Arithmetic Mean

The arithmetic mean is defined as the total sum of observations divided by the total number of observations. Here we study about the arithmetic mean of continuous series.

Calculation of arithmetic means in continuous series

When the number of items is large, we have to divide them into groups. Such groups are known as simple classes. It should be noted that there is no gap between any two successive intervals. So the data is continuous and the series is called continuous series.

Arithmetic Mean can be calculated in three different methods.

1. $$Direct \: method, \overline{X} = \frac{\sum fm}{n}$$, where m is the mid-value of class intervals.
2. $$Short \: cut \: method,\overline{X} = A + \frac{\sum fd}{n}$$, where d= X - A and A is assumed mean.
3. $$Step \: deviation \: method, \overline{X} = A + \frac{\sum fd'}{n}$$ $$\times h$$, where d' = $$\frac{d}{h}$$ and h is the length of class interval.

Arithmetic Mean can be calculated in three different methods.

1. $$Direct \: method, \overline{X} = \frac{\sum fm}{n}$$, where m is the mid-value of class intervals.
2. $$Short \: cut \: method,\overline{X} = A + \frac{\sum fd}{n}$$, where d= X - A and A is assumed mean.
3. $$Step \: deviation \: method, \overline{X} = A + \frac{\sum fd'}{n}$$ $$\times h$$, where d' = $$\frac{d}{h}$$ and h is the length of class interval.
.

#### Click on the questions below to reveal the answers

Solution:

Mean($$\overline X$$)= 5o
$$\sum fx = 750$$
N = ?

\begin{align*} Mean\:(\overline {X}) &= \frac{\sum fx}{N}\\ 50 &= \frac{750}{N}\\ or, N &= \frac{750}{50}\\ \therefore N &= 15 \: \: \: \: _{Ans} \end{align*}

Solution:

$$Mean \: (\overline{X}) =60 \\ \sum fx = 960 \\ Number \: of \: term \: (N) = ?$$

\begin{align*} Mean (\overline {X}) &= \frac{\sum fx}{N}\\ 60 &= \frac{960}{N}\\ or, N &= \frac{960}{60}\\ \therefore N &= 16 \: _{Ans} \end{align*}

Solution:

$$Mean \: (\overline{X}) =12 \\ \sum fx = 70 +10a\\ No. \: of \: term \: (N) = 5+a$$

\begin{align*}Mean \: (\overline{X}) &= \frac{\sum fx}{N}\\ 12 &= \frac{70 + 10a}{5 + a}\\ or, 60 +12a &=70 + 10a\\ or,12a -10a &= 70 - 60\\ or, 2a &= 10 \\ or, a &= \frac{10}{2}\\ \therefore a &= 5 \: \: _{Ans} \end{align*}

Solution:

$$Mean \: (\overline{X}) = 13 \\ No. \: of \: terms \: (N) = 5 + a$$

\begin{align*} Mean (\overline{X}) &= \frac{\sum X}{N} \\ or, 13 &= \frac{4 + 8+12+x+25}{5}\\ or, 65 &=49 + x\\ or, x &= 65 - 49 \\ x &= 16 \:\: _{Ans} \end{align*}

Solution:

First item mean $$(\overline {X_{1}}) = 7$$
Number of first item $$n_{1} = 4$$
Second item mean $$(\overline {X_{2}}) = 12$$
No. of Second item $$n_{2}$$= 3
Mean of 7 items $$\overline{X}= ?$$

\begin{align*} ( \overline {X}) &= \frac{n_1 \overline{X}_1 + n_2 \overline{X}_2}{n_1+n_2}\\ &= \frac{4 \times 7 + 3 \times 12}{4 + 3}\\ &= \frac{28 + 36}{7}\\ &= \frac{64}{7}\\ &= 9.14 \: \: \: _{ans} \end{align*}

$$Mean (\overline{X}) = 40 \\ No. \: of \: terms (N)= 7 \\ k = ?$$

\begin{align*} Mean(\overline{X}) &= \frac{\sum X}{N} \\ or, 40 &= \frac{10+20+30+40+50+ 60+30+k}{7}\\ or, 280 &= 240 + k\\ or, k &= 280 - 240\\ \therefore k &= 40 \: \: \: \: \: \: \: \: \: _{Ans} \end{align*}

Solution:

 Expenditure (in Rs.)x Frequency(f) fx 24 2 48 25 4 100 30 3 90 35 4 140 40 2 80

\begin{align*} Mean(\overline{X}) &= \frac{\sum fx}{N}\\ &= \frac{458}{15}\\ &= 30.53 \\ \therefore Average \: expenditure &= Rs \: 30.53 \end{align*}

Solution:

Let, the age of remaining students be x.

$$average \: age (\overline{X}) = 9 \\ Number \: of \: terms (N) = 5$$

\begin{align*} (\overline{X}) &= \frac{\sum X}{N}\\ or, 9 &= \frac{5+7+8+15+x}{5}\\ or, 45 &= 35 + x \\ \therefore x &=45 -35 = 10 \\ \: \\ \therefore & \text{The age of remaining student is 10} \end{align*}

Solution:

Calculation of mean

 Class interval cf. f mid-value (m) fm 10-20 4 4 15 60 20-30 16 12 25 300 30-40 56 40 35 1400 40-50 97 41 45 1845 50-60 124 27 55 1485 60-70 137 13 65 845 70-80 146 9 75 675 80-90 150 4 85 340 N = 150 $$\sum fm=6950$$

\begin{align*}Mean \: (\overline{X})&= \frac{\sum fm}{N}\\ &= \frac{6950}{150}\\ &= 46.33 \: _{Ans} \end{align*}

Solution:

Calculation of mean

 class interval midvalue f fm 0-10 5 5 25 10-20 15 7 105 20-30 25 8 200 30-40 35 4 140 40-50 45 6 270 N= 30 $$\sum fm = 740$$

\begin{align*} Mean (\overline{X}) &= \frac{\sum fm}{N}\\ &= \frac{740}{30}\\ &= 24.6 \: _{ans} \end{align*}

Solution:

To find the value of p.

Solution:

To find the value of p.

 Age in year No. of teachers (f) m fm 10-20 3 15 45 20-30 8 25 200 30-40 15 35 525 40-50 p 45 45p 50-60 4 55 220 N=p+30 $$\sum fm = 45p + 990$$

We know that,

\begin{align*} Mean \: (\overline {X}) &= \frac{\sum fm}{N}\\ 36 &= \frac{45p + 990}{p+30}\\ or, 1080 + 36p &= 45p +990\\ or, 45p - 36p &= 1080 -990 \\ or, 9p &= 90 \\ \therefore p &= \frac{90}{9} = 10 \: _{Ans} \end{align*}

Solution:

To find the value of p.

Solution:

To find the value of p.

 Age in year No. of teachers (f) m fm 10-20 3 15 45 20-30 8 25 200 30-40 15 35 525 40-50 p 45 45p 50-60 4 55 220 N=p+30 $$\sum fm = 45p + 990$$

We know that,

\begin{align*} Mean \: (\overline {X}) &= \frac{\sum fm}{N}\\ 36 &= \frac{45p + 990}{p+30}\\ or, 1080 + 36p &= 45p +990\\ or, 45p - 36p &= 1080 -990 \\ or, 9p &= 90 \\ \therefore p &= \frac{90}{9} = 10 \: _{Ans} \end{align*}

0%

345

152

543

251

161

321

465

785

60

60

80

20

60

80

20

60

300

750

200

700

175

80

160

120

10

7

17

20

54

92

69

22

40
85
75
55

45

21

33

65

49

59

39

69

59

69

39

49

31

40.25

61.25

35.30