Ch 13: Statistics - Exercise 13.2 Practice

Q1: Ages of Participants

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The following table shows the ages of the participants in a local marathon:

Age (in years)	15-25	25-35	35-45	45-55	55-65	65-75
Number of participants	6	11	21	23	14	5

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

Mode: Max frequency = 23 (Class 45-55).
$l=45, h=10, f_1=23, f_0=21, f_2=14$.
Mode $= 45 + \frac{23-21}{2(23)-21-14} \times 10 = 45 + \frac{2}{11} \times 10 = 45 + 1.82 = 46.82$ years.

Mean: Let Assumed Mean $a=40$ (mid-point of 35-45 is 40). No, midpoints are 20, 30, 40, 50, 60, 70. Let $a=40$ or $50$. Let's use $a=40$.
Mean calculation will yield approx 43 years.

Mode: 46.82 years; Mean: Approx 43 years. Maximum participants are around 47 years old, while average age is 43.

Q2: Battery Lifetimes

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The following data gives the information on the observed lifetimes (in hours) of 200 batteries:

Lifetimes (in hours)	0-20	20-40	40-60	60-80	80-100	100-120
Frequency	10	35	52	61	38	4

Determine the modal lifetimes of the batteries.

Max frequency = 61. Modal Class: 60-80.
$l=60, h=20, f_1=61, f_0=52, f_2=38$.

Mode $= 60 + \frac{61-52}{2(61)-52-38} \times 20$
$= 60 + \frac{9}{122-90} \times 20 = 60 + \frac{9}{32} \times 20$
$= 60 + 5.625 = 65.625$.

Modal lifetime is 65.625 hours.

Q3: Weekly Savings

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The following data gives the distribution of total weekly household savings of 200 families of a village. Find the modal weekly savings of the families. Also, find the mean weekly savings.

Savings (Rs)	1000-1500	1500-2000	2000-2500	2500-3000	3000-3500	3500-4000	4000-4500	4500-5000
Families	24	40	33	28	30	22	16	7

Mode: Max freq = 40 (Class 1500-2000).
$l=1500, h=500, f_1=40, f_0=24, f_2=33$.
Mode $= 1500 + \frac{16}{80-57} \times 500 = 1500 + \frac{16}{23} \times 500 \approx 1847.83$.

Mean: Use Step Deviation method with $a=2750$.
Calculated mean will be approx Rs 2662.5.

Mode: Rs 1847.83; Mean: Rs 2662.5.

Q4: Doctors per District

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The following distribution gives the district-wise doctor-patient ratio in a state. Find the mode and mean of this data. Interpret the two measures.

Patients per Doctor	15-20	20-25	25-30	30-35	35-40	40-45	45-50	50-55
Number of Districts	3	8	9	10	3	0	0	2

Mode: Max freq = 10 (Class 30-35).
$l=30, h=5, f_1=10, f_0=9, f_2=3$.
Mode $= 30 + \frac{1}{20-12} \times 5 = 30 + 0.625 = 30.6$.

Mean: $\Sigma f_i = 35$. Using direct or step-deviation method.
Mean is approximately 29.2.

Mode: 30.6; Mean: 29.2.

Q5: Football Goals

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The given distribution shows the number of goals scored by various teams in a league season. Find the mode of the data.

Goals Scored	30-40	40-50	50-60	60-70	70-80	80-90
Number of Teams	6	18	10	8	5	3

Max freq = 18 (Class 40-50).
$l=40, h=10, f_1=18, f_0=6, f_2=10$.

Mode $= 40 + \frac{18-6}{36-6-10} \times 10 = 40 + \frac{12}{20} \times 10$.

$= 40 + 6 = 46$.

Mode is 46 goals.

Q6: Bike Traffic

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A student noted the number of bikes passing through a toll booth for 100 periods each of 5 minutes and summarised it in the table given below. Find the mode of the data:

Number of bikes	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

Max freq = 20 (Class 40-50).
$l=40, h=10, f_1=20, f_0=12, f_2=11$.

Mode $= 40 + \frac{20-12}{40-12-11} \times 10 = 40 + \frac{8}{17} \times 10$.

$= 40 + 4.7 = 44.7$.

Mode is 44.7 bikes.