# NCERT Solutions For Class 10 Maths Chapter 14

## NCERT Solutions Class 10 Maths Statistics Types of statistics

### Ncert Solutions For Class 10 Maths Chapter 14 PDF Free Download

NCERT solutions class 10 maths chapter 14 statistics is one of the most important topics in Class 10 mathematics syllabus. Students must practice this chapter well to prepare maths more effectively. The class 10 maths chapter 14 statistics NCERT solutions is provided here to help students understand the concepts of the chapter in details. Students must practice the NCERT Solutions for class 10 maths chapter 14 thoroughly to score good marks in their class 10 board examination. Check the class 10 maths NCERT Solutions for chapter 14 pdf given below.

The topics covered in the chapter is given below

 Section Number Topic 14.1 Introduction 14.2 Mean of Grouped Data 14.3 Mode of Grouped Data 14.4 Median of Grouped Data 14.5 Graphical Representation of Cumulative Frequency Distribution 14.6 Summary

### NCERT Solutions Class 10 Maths Chapter 14 Exercises

Exercise – 1

Question 1:

A group of pupil has conducted a survey as a as the part of their college program, They collected examined data’s of plants from 20 houses and nearby areas. Find the mean of the collected plants from the given houses.

 No. of Plants Taken from 20 houses 0-2 2-4 4-6 6-8 8-10 10-12 12-14 Given number of houses 2 1 5 1 2 6 3

Name the method you use for finding the mean and why?

Solution:

 Class Interval fi xi fi xi 0-2 2 1 1 2-4 1 3 6 4-6 5 5 5 6-8 1 7 35 8-10 2 9 54 10-12 6 11 22 12-14 3 13 40 Sum fi=20 Sum fixi=163

Mean can be calculated as follows:

$\bar{x}= \frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}=\frac{163}{20}=8.15$

In this issues the $f_{i} and x_{i}$ are small so the direct method is given to solve the problem.

Question2:

The distributed wages is given for 50 workers working in the factory.

 Daily wages given to workers (in Rs) 100-120 120-140 140-160 160-180 180-200 Number of wage workers in factory 14 12 6 8 10

Find the mean of workers of daily wages given by the workers.

Solution:

In this case, value of xi is quite large and hence we should select the assumed mean method.

 Class given Interval fi xi di = xi – a fi di 100-120 14 110 -40 -480 120-140 12 130 -20 -280 140-160 6 150 0 0 160-180 8 170 20 120 180-200 10 190 40 400 Sum fi=50 Sum fi di = -240

Now, mean of deviations of daily wagesis calculated as follows:

$\bar{d}=\frac{\sum f_{i}d_{i}}{\sum f_{i}}=\frac{-240}{50}$

x = d + a = 150+(-4.8)= 145.20

Question 3:

This distribution shows that the money daily pocket given to students in the given area. The mean pocket money is Rs. 18. Find the f which is the missing frequency.

 Daily money allowance (in Rs) 11-13 13-15 15-17 17-19 19-21 21-23 23-25 Number of children 7 6 9 13 f 5 4

Solution:

 Class interval fi xi fixi 11-13 7 12 84 13-15 6 14 84 15-17 13 16 144 17-19 9 18 234 19-21 f 20 20f 21-23 4 22 110 23-25 5 24 96 Sum fi=44+f Sum fixi= 752+20f

We have;

$\bar{x}= \frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}$ $18= \frac{752+20f}{44+ f_{i}}$

18(44+18f) = 752+20f

792+18f = 752+20f

2f = 40

Missing frequency f= 20

Question4:

The number of 30 women were checked by one doctor and the distributed the heartbeat of them in following distribution table. Find the mean of heart beats of the given per minute for these thirty women, by choosing a suitable following method.

 Heartbeat per minute in given number 65-68 68-71 71-74 74-77 77-80 80-83 83-86 Number of 30 women 2 3 4 7 8 2 4

Solution:

 Class Interval fi xi di = xi – a fi di 65-68 2 66.5 -9 -18 68-71 3 69.5 -6 -24 71-74 4 72.5 -3 -9 74-77 7 75.5 0 0 77-80 8 78.5 3 21 80-83 2 81.5 6 24 83-86 4 84.5 9 18 Sum fi= 30 Sum fi di = 12

Now, mean can be calculated as follows:

$\bar{d}=\frac{\sum f_{i}d_{i}}{\sum f_{i}}=\frac{12}{30}=0.4$ $\bar{x}=\bar{d}+a=0.4+75.5=75.9$

Question 5:

The oranges were packed and were sold by the fruit seller in a  market. The numbers of oranges are arranged in different boxes by the fruit seller. Through the number of oranges the number of oranges are distributed in following manner.

 No.of oranges 50-52 53-55 56-58 89-61 62-64 No. of boxes 15 110 135 115 25

In the distributed pattern the number of oranges. Find the mean. Choose the method to find the mean:

Solution:

 Class interval fi xi di=x-a fi di 50-52 110 54 -6 90 53-55 15 51 -3 -330 56-58 135 60 0 0 59-61 25 57 3 345 62-64 115 63 6 150 Sum fi= 400 Sum fi di=75

Mean can be calculated as follows:

$\bar{d}=\frac{\Sigma f_{i}d_{i}}{\Sigma f_{i}}=\frac{75}{400}=0.1875$ $\bar{x}=\bar{d}+a=0.1875+57=57.1875$

In this case, there are wide variations in $f_{i}$ and hence assumed mean method is used.

Question 6:

The following distribution shows the food expenditure of the given households from the locality:

 Daily food expenditure (in Rs) 50-100 100-150 150-200 200-250 250-300 Number of given households 4 5 12 2 2

Find the mean daily expenditure on food by a suitable method.

Solution:

 Class Interval fi xi di= xi – a ui = di/h fiui 50-100 4 125 -100 -2 -8 100-150 5 175 -50 -1 -5 150-200 12 225 0 0 0 200-250 2 275 50 1 2 250-300 2 325 100 2 4 Sum fi = 25 Sum fiui = -7

Mean can be calculated as follows:

$\bar{x}=a+\frac{\sum f_{i}u_{i}}{\sum f_{i}}\times h$ $=225+\frac{-7}{25}\times 50 =225$

Question 7:

To find concentration in sulphur dioxide (in parts per million, i.e. ppm),The parts of data are collected by different areas from cities are given in distributed table:

 Concentration in SO2 (in ppm) Frequency of F 0.00-0.04 4 0.04-0.08 9 0.08-0.12 9 0.12-0.16 2 0.16-0.20 4 0.20-0.24 2

Give the mean of the sulphur dioxide collected from the given cities.

Solution:

 Class Interval fi xi fixi 0.00-0.04 9 0.02 0.08 0.04-0.08 4 0.06 0.54 0.08-0.12 9 0.10 0.90 0.12-0.16 2 0.14 0.28 0.16-0.20 2 0.18 0.72 0.20-0.24 4 0.22 0.44 Sum fi=30 Sum fixi=2.96

Mean can be calculated as follows:

$\bar{x}=\frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}=\frac{2.96}{30}=0.099 ppm$

Question 8:

The absentee record of 40 students is handled by the teacher for one complete term. Find the mean of absent days of the given students:

 Number of following days 0-6 6-10 10-14 14-20 20-28 28-38 38-40 Number of following students 10 11 4 7 3 4 1

Solution:

 Class Interval fi xi fixi 0-6 11 3 33 6-10 10 8 80 10-14 4 12 84 14-20 7 17 68 20-28 3 24 96 28-38 4 33 99 38-40 1 39 39 Sum fi = 40 Sum fixi= 499

Mean can be calculated as follows:

$\bar{x}=\frac{\sum f_{i}x_{i}}{\sum f_{i}}$= $\frac{499}{40}$=12.4

Question 9:

The data of the literacy digit from the 35 cities is given in the following table. Find the mean of the 35 literacy cities.

 Literacy rate from 35cities (in %) 40-45 50-55 60-65 70-75 80-85 Cities in values 10 3 8 11 3

Solution:

 Class Interval fi xi di=xi-a ui=di/h fiui 40-45 10 50 -20 -2 -6 50-55 3 60 -10 -1 -10 60-65 8 70 0 0 0 70-75 11 80 10 1 8 80-85 3 90 20 2 6 Sum fi=35 Sum fiui= -2

Mean can be calculated as follows:

$\bar{x}=a+\frac{\Sigma f_{i}u_{i}}{\Sigma f_{i}}\times h$ $= 70+\frac{-2}{35}\times 10=69.42$

EXERCISE – 2

Question 1:

The age of patients admitted in a hospitals are given in the following table for a complete year.

 Age (in years) 5-15 15-25 25-35 35-45 45-55 55-65 Number of patients 6 11 21 23 14 5

Calculate the mode and mean of the data above. Compare and interpret the two measures of central tendency.

Solution:

Modal class = 35-45, length = 35, height = 10, $a_{1} = 23, a_{0} = 21\, and\, a_{3} = 14$

Therefore, Mode = $l +\frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}}\times h$

= $35+\left ( \frac{23-21}{2\times 23-21-14} \right )\times 10$

= $35+\frac{2}{11}\times 10=36.8$

Calculating Mean

 CLASS INTERVAL ai zi aizi 5-15 6 10 60 15-25 11 20 220 25-35 21 30 630 35-45 21 30 920 45-55 14 50 700 55-65 5 60 300 Sum ai = 80 Sum aizi= 2830

$\bar{p}=\frac{\sum a_{i}x_{i}}{\sum a_{i}}=\frac{2830}{80}=35.37$

$\ therefore$The mode of data displays that the maximum number of patients is in the age group of 26.8, whereasthe average age of all the patients is 35.37.

Question 2:

The data given below provides the information for the observed lifetime (in hours) of 225 electrical components:

 Lifetime ( in hours) 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100  – 120 Frequency 10 35 52 61 38 29

Find the modal lifetimes of the components.

Solution:

Modal class =  60-80, l = 60,$a_{1} = 61, a_{0} = 52, a_{2} = 38$ and h = 20

$Mode = l + \left ( \frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}} \right )\times h$

$= 60 + \left ( \frac{61 – 52}{2 \times 61 – 52 – 38 } \right ) \times 20$

$= 60 + \frac{9}{32} \times 20 = 65.62$

Question 3:

The total monthly household expenses of 200 families of a town are given in the following table. Calculate the modal monthly expenses of the families.  Also, calculate the mean monthly expenses.

 Expenses Number of families 1000 – 1500 24 1500 – 2000 40 2000 – 2500 33 2500 – 3000 28 3000 – 3500 30 3500 – 4000 22 4000 – 4500 16 4500 – 5000 7

Solution:

Modal class = 1500-2000, l = 1500,$a_{1} = 40, a_{0} = 24, a_{2} = 33$ and h = 500

$Mode = l + \left ( \frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}} \right )\times h$

$= 1500 + \left ( \frac{40 – 24}{2 \times 40 – 24 – 33} \right ) \times 500$

$= 1500 + \left ( \frac{16}{23} \right ) \times 500 = 1847.82$

Calculation of mean:

 Class Interval ai zi di =zi – b ui= di/h aiui 1000 – 1500 24 1250 -1500 -3 -72 1500 – 2000 40 1750 -1000 -2 -80 2000 – 2500 33 2250 -500 -1 -33 2500 – 3000 28 2750 0 0 0 3000 – 3500 30 3250 500 1 30 3500 – 4000 22 3750 1000 2 44 4000 – 4500 16 4250 1500 3 48 4500 – 5000 7 4750 2000 4 28 ai = 200 aiui = -35

$\bar{p}=\frac{\sum a_{i}x_{i}}{\sum a{i}}\times h$ $= 2750 – \frac{35}{200} \times 500$

= 2750 – 87.5 = 2662.50

Question 4:

The state – wise teacher- student ratio in a college in India are given in the following table. Calculate the mean and mode of the given data. Interpret the two measures.

 Number of students per teacher Number of states/UT 15 – 20 3 20 – 25 8 25 – 30 9 30 – 35 10 35 – 40 3 40 – 45 0 45 – 50 0 50 – 55 2

Solution:

Modal class = 30-35, l = 30, $a_{1} = 10, a_{0} = 9, a_{2} = 3$ and h = 5

$Mode = l +\left ( \frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}} \right )\times h$

$= 30 – \left ( \frac{20 – 9}{2 \times 10 – 9 – 3 } \right ) \times 5$

$= 30 – \frac{1}{8} \times 5 = 30.625$

Calculating mean:

 Number of students per teacher ai zi di= zi –b ui = ui/h aiui 15 – 20 3 17.5 -15 -3 -9 20 – 25 8 22.5 -10 -2 -16 25 – 30 9 27.5 -5 -1 -9 30 – 35 10 32.5 0 0 0 35 – 40 3 37.5 5 1 3 40 – 45 0 42.5 10 2 0 45 – 50 0 47.5 15 3 0 50 – 55 2 52.5 20 4 8 Sum ai = 35 sum aiui= -23

$\bar{p}=\frac{\sum a_{i}x_{i}}{\sum a_{i}}\times h$

$= 30.5 – \frac{23}{35}\times 5$

$= 30.5 – \frac{23}{7} = 29.22$

Therefore, from the mode we know that the maximum number of states has 30 – 35 students per teacher, and from the mean we know that the average ratio of students per teacher is 29.22

Question 5:

Provided table shows the total runs scored by some of the top batsman of the world in one-day cricket matches.

 Runs Scored Number of Batsman 3000 – 4000 4 4000 – 5000 18 5000 – 6000 9 6000 – 7000 7 7000 – 8000 6 8000 – 9000 3 9000 – 10000 1 10000 – 11000 1

Calculate the mode of the given information.

Solution:

Modal class = 4000-5000, l = 4000, $a_{1} = 18, a_{0} = 4, a_{2} = 9$ and h = 1000

$Mode = l + \left ( \frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}} \right )\times h$ $= 4000 + \left ( \frac{18 – 4}{2 \times 18 – 4 -9} \right )\times 1000$ $= 4000 + \left ( \frac{14}{23} \right )\times 1000 = 4608.70$

Question 6:

A complete observation of the number of cars on the roads for 100 periods of 3 minutes are summarized in the following table. Calculate the mode of the data.

 Number of cars 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60 60 – 70 70 – 80 Frequency 7 14 13 12 20 11 15 8

Solution:

Modal class = 40 – 50, l = 40, $a_{1} = 20, a_{0} = 12, a_{2} = 11$ and h = 10

$Mode = l + \left ( \frac{a_{1}-a_{0}}{2a_{1}-a_{0}-a_{2}} \right )\times h$

$= 40 + \left ( \frac{20 – 12}{2 \times 20 – 12 – 11} \right )\times 10$

$= 40 + \left ( \frac{8}{17} \right )\times 10 = 44.70$

Exercise 3

Question 1:

The following frequency distribution gives the monthly consumption of an electricity of 68 consumers in a locality. Find the median, mean and mode of the data and compare them.

 Monthly consumption(in units) No. of customers 65-85 4 85-105 5 105-125 13 125-145 20 145-165 14 165-185 8 185-205 4

Solution:

 Class Interval Frequency Cumulative frequency 65-85 4 4 85-105 5 9 105-125 13 22 125-145 20 42 145-165 14 56 165-185 8 64 185-205 4 68 N=68

Where, n = 68 and hence $\frac{n}{2}=34$

Hence, the median class is 125-145 with cumulative frequency = 42

Where, l = 125, n = 68, cf = 22, f = 20, h = 20

Median is calculated as follows:

Median=$l+\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$

=$125+\left ( \frac{34-22}{20} \right )X20$

=125+12=137

Mode = Modal class = 125-145, $f_{1} = 20, f_{0} = 13, f_{2} = 14$ & h = 20

Mode=$l+\left ( \frac{f_{1}-f_{0}}{2Xf_{1}-f_{0}-f_{2}} \right )Xh$

=$125+\left ( \frac{20-13}{2X20-13-14} \right )X20$

=$125+\frac{7}{13}X20$

=125+10.77

=135.77

Calculate the Mean:

 Class Interval fi xi di=xi-a ui=di/h fiui 65-85 4 75 -60 -3 -12 85-105 5 95 -40 -2 -10 105-125 13 115 -20 -1 -13 125-145 20 135 0 0 0 145-165 14 155 20 1 14 165-185 8 175 40 2 16 185-205 4 195 60 3 12 Sum fi= 68 Sum fiui = 7

$\bar{x}=a+\frac{\sum f_{i}u_{i}}{\sum f_{i}}X h$ $=135+\frac{7}{68}X20$

=137.05

Mean, median and mode are more/less equal in this distribution.

Question 2:

If the median of a distribution given below is 28.5 then, find the value of an x &y.

 Class Interval Frequency 0-10 5 10-20 x 20-30 20 30-40 15 40-50 y 50-60 5 Total 60

Solution:

n = 60

Where,$\frac{n}{2}$ = 30

Median class is 20 – 30 with a cumulative frequency = 25 + x

Lower limit of median class = 20, cf = 5 + x , f = 20 & h = 10

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$

Or,28.5=20+$\left ( \frac{30-5-x}{20} \right )X10$

Or,$\frac{25-x}{2}$

Or,   25-x=17

Or,    x= 25 – 17   =8

Now, from cumulative frequency, we can identify the value of x + y as follows:

60=5+20+15+5+x+y

Or, 45+x+y=60

Or, x + y=60-45=15

Hence, y = 15 – x= 15 – 8 = 7

Hence, x = 8  &  y = 7

Question 3:

The Life insurance agent found the following data for the distribution of ages of 100 policy holders. Calculate the median age, if policies are given only to the  persons whose age is 18 years onwards but less than the 60 years.

 Age(in years) Number of policy holder Below 20 2 Below 25 6 Below 30 24 Below 35 45 Below 40 78 Below 45 89 Below 50 92 Below 55 98 Below 60 100

Solution:

 Class interval Frequency Cumulative frequency 15-20 2 2 20-25 4 6 25-30 18 24 30-35 21 45 35-40 33 78 40-45 11 89 45-50 3 92 50-55 6 98 55-60 2 100

Where, n = 100 and $\frac{n}{2}$ = 50

Median class = 35-45

Then, l = 35, cf = 45, f = 33 & h = 5

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$ $=35+\left ( \frac{50-45}{33} \right )X5$ $=35+\left ( \frac{25}{33} \right )$

=35.75

Question 4:

The lengths of 40 leaves in a plant are measured correctly to the nearest millimetre, and the data obtained is represented as in the following table:

 Length(in mm) Number of leaves 118-126 3 127-135 5 136-144 9 145-153 12 154-162 5 163-171 4 172-180 2

Find the median length of leaves.

Solution:

 Class Interval Frequency Cumulative frequency 117.5-126.5 3 3 126.5-135.5 5 8 135.5-144.5 9 17 144.5-153.5 12 29 153.5-162.5 5 34 162.5-171.5 4 38 171.5-180.5 2 40

Where, n = 40 and $\frac{n}{2}$ = 20

Median class = 144.5-153.5

then, l = 144.5, cf = 17, f = 12 & h = 9

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$ $=144.5+\left ( \frac{20-17}{12} \right )X9$ $=144.5+\left ( \frac{9}{4} \right )$

=146.75

Question 5:

The following table gives the distribution of a life time of 400 neon lamps.

 Lifetime(in hours) Number of lamps 1500-2000 14 2000-2500 56 2500-3000 60 3000-3500 86 3500-4000 74 4000-4500 62 4500-5000 48

Find the median lifetime of a lamp.

Solution:

 Class Interval Frequency Cumulative 1500-2000 14 14 2000-2500 56 70 2500-3000 60 130 3000-3500 86 216 3500-4000 74 290 4000-4500 62 352 4500-5000 48 400

Where, n = 400 &$\frac{n}{2}$ = 200

Median class = 3000 – 3500

Therefore, l = 3000, cf = 130, f = 86 & h = 500

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$ $=3000+\left ( \frac{200-130}{86} \right )X500$

=3000+406.97

=3406.97

Question 6:

In this 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in English alphabets in the surnames was obtained as follows:

 Number of letters 1-4 4-7 7-10 10-13 13-16 16-19 Number of surnames 6 30 40 16 4 4

Determine the number of median letters in the surnames. Find the number of mean letters in the surnames and also, find the size of modal in the surnames.

Solution: To Calculate median:

 Class Interval Frequency Cumulative Frequency 1-4 6 6 4-7 30 36 7-10 40 76 10-13 16 92 13-16 4 96 16-19 4 100

Where, n = 100 &$\frac{n}{2}$ = 50

Median class = 7-10

Therefore, l = 7, cf = 36, f = 40 & h = 3

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$ $=7+\left ( \frac{50-36}{40} \right )X3$ $=7+\left ( \frac{14}{40} \right )X3$

=8.05

Calculate the Mode:

Modal class = 7-10,

Where, l = 7, f1 = 40, f0 = 30, f2 = 16 & h = 3

Mode=$l+\left ( \frac{f_{1}-f_{0}}{2Xf_{1}-f_{0}-f_{2}} \right )Xh$

=$7+\left ( \frac{40-30}{2X40-30-16} \right )X3$

=$7+\frac{10}{34}X3$

=7.88

Calculate the Mean:

 Class Interval fi xi fixi 1-4 6 2.5 15 4-7 30 5.5 165 7-10 40 8.5 340 10-13 16 11.5 184 13-16 4 14.5 51 16-19 4 17.5 70 Sum fi = 100 Sum fixi = 825

$\bar{x}=a+\frac{\sum f_{i}u_{i}}{\sum f_{i}}X h$ $=\frac{825}{100}$

=8.25

Q7. The distributions of below gives a weight of 30 students of a class. Find the median weight of a students.

 Weight(in kg) 40-45 45-50 50-55 55-60 60-65 65-70 70-75 Number of students 2 3 8 6 6 3 2

Solution:

 Class Interval Frequency Cumulative frequency 40-45 2 2 45-50 3 5 50-55 8 13 55-60 6 19 60-65 6 25 65-70 3 28 70-75 2 30

Where, n = 30 and $\frac{n}{2}$= 15

median class = 55-60

therefore, l = 55, cf = 13, f = 6 & h = 5

Median=l+$\left ( \frac{\frac{n}{2}-cf}{f} \right )Xh$ $=55+\left ( \frac{15-13}{6} \right )X5$ $=55+\left ( \frac{5}{3} \right )$

=56.67

Question 7:

The distribution shows the pocket money of students in an area. The mean pocket money is Rs 18. Find the missing frequency f.

 Daily pocket allowance (in Rs) 11-13 13-15 15-17 17-19 19-21 21-23 23-25 Number of children 07 06 09 013 F 05 04

Solution:

 Class interval fi xi fixi 11-13 07 012 84 13-15 06 014 84 15-17 09 016 144 17-19 013 018 234 19-21 f 020 20f 21-23 05 022 110 23-25 04 024 96 Sum fi = 44+f Sum fixi = 752+20f

We have,

Mean pocket money, $x_{i}$ = 18(given)

Hence substituting for fixi,

18(44+f) = 752 + 20f

Or, 792 + 18f = 752 + 20f

Or, 2f = 40

$\rightarrow f=20$

The modern world is highly data-oriented and it becomes necessary to know to represent data in a meaningful way. The branch of mathematics which deals with the representation of data in a meaningful way is known as statistics. Let us look at a few examples of how statistics is useful in our daily life

• The data that you see in the weather forecast section of the news is due to the goodwill of statistics.
• The selection of a player in any sport is mostly made based on the player’s statistical performance report.

In this chapter, students are introduced to histograms, graphical representations of data in the form of bar charts, cumulative frequency tables and frequency polygons. Students will be introduced to the three measures of central tendency. The three measures of central tendency are mean, median and mode.

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