Question

The table shows the daily sales (in $1000) of shopping mall for some randomly selected
days
Sales 1.1-1.5 1.6-2.0 2.1-2.5 2.6-3.0 3.1-3.5 3.6-4.0 4.1-4.5
Days 18 27 31 40 56 55 23
Use it to answer questions 13 and 14.
13. What is the approximate value for the modal daily sales?
A. $3,129.41 B. $2,629.41 C. $3,079.41 D. $3,123.53
14. The approximate median daily sales is …
A. $3,130.36 B. $2,680.36 C. $3,180.36 D. $2,664

Answer 1

Classes	Frequency (f)
1.1-1.5	18
1.6-2.0	27
2.1-2.5	31
2.6-3.0	40
3.1-3.5	56
3.6-4.0	55
4.1-4.5	23

First we will convert our inclusive frequency distribution to exclusive

Class	Frequency
1.05 - 1.55	18
1.55 - 2.05	27
2.05 - 2.55	31
2.55 - 3.05	40
3.05 - 3.55	56
3.55 - 4.05	55
4.05 - 4.55	23

Solution:

Class	Frequency (f)	cf
1.05-1.55	18	18
1.55-2.05	27	45
2.05-2.55	31	76
2.55-3.05	40	116
3.05-3.55	56	172
3.55-4.05	55	227
4.05-4.55	23	250
---	---	---
--	n=250	--

14

To find Median Class
= value of (n / 2)th observation

= value of (250 / 2)th observation

= value of 125th observation

From the column of cumulative frequency cf,

we find that the 125th observation lies in the class 3.05-3.55.

∴ The median class is 3.05-3.55.

Now,

∴L=lower boundary point of median class =3.05

∴n=Total frequency =250

∴cf=Cumulative frequency of the class preceding the median class =116

∴f=Frequency of the median class =56

∴c=class length of median class =0.5

Median M=L+n2-cff⋅c

=3.05+125-11656⋅0.5

=3.05+956⋅0.5

=3.05+0.08036

=3.13036

hence median sales = 3130.36

13.

Similarly modal daily sales =

we can estimate the Mode using the following formula:

Estimated Mode =

L + fm − fm-1(fm − fm-1) + (fm − fm+1) × w

where:

• L is the lower class boundary of the modal group
• fm-1 is the frequency of the group before the modal group
• fm is the frequency of the modal group
• fm+1 is the frequency of the group after the modal group
• w is the group width

now

We can easily find the modal group (the group with the highest frequency), which is 3.05 -3.55

L = 3.05

fm-1 =40

fm =56

fm+1 = 55

w = 0.5

mode =

mode = 3.05 + 0.4705 = 3.5205

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