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Introduction To Statistics: Populations And Samples

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Question:

Describe about the Population and Samples in form of Statistics?

 

Answer:

i) Data for the amount paid for the sample of families appears to follow normal distribution with positively skewed or skewed to the right data. It appears to be unimodal with peak observed around the category ‘350 but under 400’.
 
ii) Histogram –

Amount Paid

Data for the amount paid for the sample of families appears to follow normal distribution with positively skewed or skewed to the right data. It appears to be unimodal with peak observed around the category ‘350 but under 400’.

b) Nominal data – Data which can only be categorised. Even if it contains numerical data no numerical operations can be done. Eg: Serial No., Name of the earning person of the family.

Ordinal data – The data can be ordered in increasing or decreasing order. Eg: amount paid for the sample of families are ordered in increasing order of amount.

Sample data – consists of few observations drawn from the population. More than one sample can be drawn from the population. Eg: amount paid for the sample of families for 50 families included.

Population data – Consists of each and every observation.

a) Ogive is a line histogram plotting cumulative frequency against classes.

Frequency Table-

Class

Frequency

Cumulative Frequency

Under 250

 

0

250 but under 300

8

8

300 but under 350

10

18

350 but under 400

14

32

400 but under 450

9

41

450 but under 500

6

47

500 but under 550

1

48

550 but under 600

2

50

 

Ogive graph –

Ogive graph

 

For the above grouped data summary statistics was calculated including mean, median, mode and standard deviation. Standard Deviation and variance provides measure of dispersion.

Mean

381

Median

375

Mode

375

SD

75.92

Variance

5764


b) The mean obtained by colleague having access to raw data is £370 and mean calculated from grouped data is £381, which is little higher than the actual mean.

Eg: Suppose below values present the number of different playing cards owned by group of friends –

9, 15, 11, 12, 3, 5, 10, 20, 14, 6, 8, 8, 12, 12, 18, 15, 6, 9, 18, 11

Average for this data will be given by –

Mean =  sum/N = (9+15+11+12+3+5+10+20+14+6+8+8+12+12+18+15+6+9+18+11)/20= 11.1

Now, if we construct its frequency table and think that we lost the original data then,

Number of Cards

Midpoint

Frequency

Product

1 - 5

3

2

6

6 - 10

8

7

56

11 - 15

13

8

104

16 - 20

18

3

54

Sum

 

20

220

Mean

 

 

11


Mean = sum of product/sum of frequency = 220/22 = 11

So, actual mean is 11.1 and grouped frequency mean is 11.

Difference between mean calculated using the two above different methods may be due to rounding of data because in grouped data we are estimating mean based on midpoint of the class interval and frequency and not the actual data.

i) Mean is obtained by dividing sum of all observations with the total number of observations to get the midpoint of the data while the median is the middle value which separates data into two equal half.

Mean gets influenced by few higher values while median does not. In our case, Amount paid above £450 are very less and will influence mean but not median. So, median will provide better measure for central tendency.

ii) Approximately 82% people paid less than £475 and 60% of people pay less than or equal to £375 which is the midpoint of class £350 but under £400. These values are derived by calculating cumulative percent.

Ogive for Cumulative Percent

Ogive for Cumulative Percent


iii) To calculate 68% interval of amount paid we add and subtract standard deviation from mean which is given by –

Lower 68%  = mean – sd = 381 – 75.92 = 305.08

Upper 68% = mean + sd = 381+75.92 = 456.92

Thus, 68% interval is £305.08 - £456.92.

i) Graph –

Number of Families visiting theme park


Based on the equation obtained forecast values for four quarters of 2015 are –

2015

     

1

2

3

4

408.0

412.8

417.7

422.5


b) These values are not forecast as the seasonal effect observed in the values from 2011 to 2014 is missing in the trend values for 2015.

c) As the values are discrete and grouped in the three categories for four quarters column graph was used to represent the data. Also, direct comparison in each quarter can be made.

Types of Visitors to Theme Park

 

References:

1) Populations and Samples. Stat Trek Teach yourself statistics. Available from: https://stattrek.com/sampling/populations-and-samples.aspx (accessed on 11 Mar 2015)

2) Introduction to Statistics. Available from: https://www.sagepub.com/upm-data/42772_1_Introduction_to_Statistics.pdf (accessed on 11 Mar 2015)

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