Chapter 3 - Numerical Description Statistis

Statistcs

1. NUMERICAL DESCRIPTIVE
MEASURES
Chapter 3

2. Descriptive Statistics - tools used to
summarize and present the gathered data in
graphical, tabular, or numerical form so readers can
easily understand it.

3. I. Measures of Central Tendency
•Mean - Average = Sum of the values/number of values
•Median - Middle value in an ordered array = (n + 1)/2 ranked value
•Mode - most frequently occurring value

4. MEAN
MEDIA
N
MODE
Lee’s quiz
scores:
60 70 70 70 80
Pat’s quiz
scores:
45 45 70 90 100
Sam’s quiz
scores:
50 60 70 80 90
Xiao’s quiz
scores:
50 50 70 90 90

5. II. Measures of Variation
•Range = Xlargest - Xsmallest
•Sample Variance = S^2 is the sum of the squared differences around the mean
divided by the sample size minus 1
•Sample Standard deviation = S = SQRT(S^2) is the square root of the sum of the
squared differences around the mean divided by the sample size minus 1
•Coefficient of Variation = CV=(S/X-bar)100% is equal to the standard deviation
divided by the mean multiplied by 100%
•Z score = (X - X-bar)/S is equal to the difference between the value and the mean,
divided by the standard deviation. It identifies outliers for values < -3 or >+3

6. III. Shape or Pattern
•Skewness measures the extent to which the data values are not
symmetrical around the mean
•Mean < Median: negative or left-skewed distribution
•Mean = Median: symmetrical distribution or zero skewness
•Mean > Median: positive or right-skewed distribution

7. IV. Exploring Numerical Variables
•Quartiles split the values into four equal parts
•Q1 or the first quartile ; smallest 25% of the values vs largest 75%
•Q1 = (n + 1)/4 ranked value
•Q2 or the median; 50% of the values are smaller than or equal to the median and
50% are larger than or equal to the median
•Q3 or third quartile; smallest 75% of the values vs largest 25%
•Q3 = 3(n+1)/4 ranked value
•INTERQUARTILE RANGE or the Midspread = Q3 - Q1

8. •Five-number summary consists of the smallest value (Xsmallest), the
first quartile (Q1), the median, the third quartile (Q3), and the largest
value (Xlargest)
•Xsmallest Q1 Median Q3 Xlargest

9. •The Boxplot visualizes the shape of the distribution of the values for a
variable
Q1. MEDIAN. Q3
3 4 5 6 7 8 9 10 11 12 13

11. 3.1
For a sample of data where n=6
given below:
X X-bar (X-Xbar)^2 Z score
8 5 10 7 3 6
3 5 6 7 8 10
ORDERED ARRAY
Mean
Median
Mode
Range
Variance
Std dev.
CV
Z scores
Shape: Skewness
Q1
(n+1)/4
Q3
3(n+1)/4
IQR
FIVE-NUMBER SUMMARY
BOXPLOT using the Five-Number Summary

12. 3.1For a sample of data where n=6
given below:
X X-bar(X-
Xbar)^2
Z score
8 5 10 7 3 6 3 6.512.25 -1.44
3 5 6 7 8 10 5 6.5 2.25 -4.01
6 6.5 0.25 -0.21
Mean 3939/6 6.5 7 6.5 0.25 0.21
Median 6.5 8 6.5 2.25 0.62
Mode None 10 6.512.25 1.44
Range 10-3 7 29.5
Variance 29.5/5 5.9
Std dev. SQRT5.9 2.43
CV 2.43/6.5)*10
0
37.38%
Z scores Lowest -1.44
Highest 1.44Therefore, no outlier bec no score lower than -3 and no score more than +3
Shape:
Skewness
Mean=Medi
an
Symmetrical distribution (zero skewness)
Q1 1.75 2nd ranked value = 5
Q3 5.25 5th ranked value = 8
IQR 8-5=3
FIVE-NUMBER
SUMMARY
3 5. 6.5. 8. 10
Q1. MEDIAN. Q3
3 4 5 6 7 8 9 10

13. 3.2For a sample of data where
n=7 given below:
X X-bar(X-Xbar)^2Z score
8 3 10 6 4 13 5
Mean
Median
Mode
Range
Variance or
S^2
Std dev. Or S
CV
Z scores Lowest
Highest
Shape:
Skewness
Q1
Q3
IQR
FIVE-NUMBER SUMMARY
BOXPLOT using the Five-
Number Summary

14. 3.10The following is the overall download and upload speeds in mbps for nine carriers in the USA.
Carrier Download speed Upload speed
Verizon 24 14.3
T-mobile 22.7 13.2
AT&T 20.8 9.1
Metro PCS 16.7 11.1
Sprint 11.2 6.4
Virgin mobile 10.8 6.2
Boost 10.3 6
Straight Talk 7.1 3
Cricket 4.5 3.8
Mean
Median
Mode
Range
Variance
Std dev.
CV
Z scores
Shape: Skewness
Q1 (n+1)/4
Q3 3(n+1)/4
IQR
FIVE-NUMBER SUMMARY
BOXPLOT using the Five-Number Summary
What conclusions can you reach concerning the download and upload speed of various carriers.