Statistics Descriptive for Group Data

Probability Theory

Grouped Data

R. Fajriyah¹

1Dept of Statistics UII Jogjakarta, Indonesia

April 7, 2015

Outline

1 How to construct histogram

2 How to construct Boxplot

3 Mean, Median and Mode from grouped frequencies

4 Variance from grouped frequencies

Data

Here is the data on starting salaries of 1995 Psychology graduates. When constructing a histogram it is helpful to sort the observations. 08820 10800 12000 12500 13000 14000 15000 16000 16500 16600 16700 16900 16900 17000 17000 17600 17880 18000 18000 18000 18000 18000 18000 18000 18000 18000 18000 18500 18680 19100 20000 20000 20000 20000 20000 20300 20900 22000 23000 23000 23000 23000 23400 24000 25000 25000 26000 26000 27000 30000 30000 32500 37000 48000

Steps to construct histogram 1

1 Compute the range : Maximum-Minimum. Minimum = 8820 Maximum = 48000 Range = 39180

2 Decide how many intervals you would like. Use the square root of the number of observations (after rounding). Here, that is the square root of 54 = 7.34; round up and use 8.

3 Compute the interval width: Range/# Intervals = 39180/8 = 4897.5; round up to 5000.

4 Start the first interval at a convenient value below the minimum. Here the minimum is 8820, so begin at 7500 (other choices are equally acceptable).

5 The intervals then begin at 7500 and have a width of 5000.

So, the first interval runs from 7500 to 12500 and so on.

Steps to construct histogram 2

1 Construct the dot plot to see many intervals you would like.

2 Compute the interval width: Range/# Intervals ; round up to the same scale as the original data.

3 Start the first interval at a convenient value below the minimum.

Steps

1 Draw a number line that includes the range of observations. ComputeQ₁,Q₂, andQ₃.

2 Above the line drawn in the first step, draw a box extending fromQ₁toQ₃. Inside the box, draw a line at the median (Q₂).IQR =Q₃−Q₁

3 To identify the outliers, compute the lower and upper fences: f_L=Q₁−1.5∗(IQR)and the upper fence is f_U =Q₃+1.5∗(IQR)

4 Observations located beyond the fence are classified as outliers and are identified with an asterisk (*).

5 If there are no outliers, extend horizontal line segments (whiskers) from the ends of the box to the smallest and largest observations. If there are outliers, extend the whiskers to the smallest and largest non-outliers.

Data

9, 15, 11, 12, 3, 5, 10, 20, 14, 6, 8, 8, 12, 12, 18, 15, 6, 9, 18, 11 Please compute: Mean, Median, Mode

Table

Number of games Frequency

1-5 2

6-10 7

11-15 8

16-20 3

Computation I

For grouped data, we cannot find the exact Mean, Median and Mode, we can only give estimates.

To estimate the Mean use the midpoints of the class intervals.

To Estimate MedianM_gd =L+

n 2−cf_b

fm w, where

Lis the lower class boundary of the group containing the median

nis the total number of data

cfbis the cumulatif frequency of the groups before the median group

f_mis the frequency of the median group wis the group width

To Estimate ModeM_gd =L+ ^fm−fm−1 (^fm−fm−1)^+fm−fm+1

w, where

Computation II

Lis the lower class boundary of the modal group

fm−1is the frequency of the group before the modal group f_mis the frequency of the modal group

f_m+1is the frequency of the group after the modal group wis the group width

For continuous data use limits (rather than boundaries) for median and mode

Results

Our final result is

Estimated Mean: 11 Estimated Median: 11.625 Estimated Mode: 11.833..

Another example

Length Frequency

150 - 154 5

155 -159 2

160 - 164 6

165 - 169 8

170 - 174 9

175 - 179 11

180 - 184 6

185 - 189 3

Another example

Age Number 0 - 9 20 10 - 19 21 20 - 29 23 30 - 39 16 40 - 49 11 50 - 59 10 60 - 69 7 70 - 79 3 80 - 89 1

midpoints: 5, 15, 25, etc

Steps

s²=

P(xi−¯x)²fi

Pfi−1

or

s²=

Px_i²f_i−(^P_{xi fi})² P fi

f_i

Thank you for your attention!

References I

Walpole, Myers, Myers, and Ye, 2011.Probability and Statistics for Engineers and Scientists, ninth edition, Prentice Hall, Boston