The document discusses descriptive statistics, detailing measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation). It provides formulas and examples for calculating these statistics for both ungrouped and grouped data. Additionally, it explains the computations involved in deriving standard deviation using two methods: long method and coded deviation method.

1. ADVANCE STATISTICS
Jemer M. Mabazza, LLB, PhD
Graduate School
Mallig Plains Colleges

2. DESCRIPTIVE STATISTICS
 Descriptive Statistics
• Two main function of descriptive statistics
1. To carry out the measure of central tendency
2. To calculate a measure of dispersion or variability
• Measure of Central Tendency
1. Mean. Refers to descriptive statistic that summarizes the centre of a range of
values. It is computed by mean of adding all the scores or values in each
condition and dividing them by the number of scores or values.
2. Median. Refers to descriptive statistic that summarizes the middle of a range of
values after arranging items from highest to lowest or vice versa.
3. Mode. Refers to the descriptive statistics that identifies the value with the higher
frequency.

3. 10 13 13 22 23 25 30 35 40
• Example
Mean =
10+ 13+13+22+23+25+30+35+40
9
=
211
9
= 23.44
Median = 23
• RULES:
 If N is an ODD NUMBERS, get the EXACT MIDDLE SCORE/NUMBER
 If N is an EVEN NUMBERS, get the AVERAGE OF TWO MIDDLE
SCORES/NUMBERS
Mode = 13
DESCRIPTIVE STATISTICS

4. • Illustration of the Rules in the Median
 If N is an ODD NUMBERS, get the EXACT MIDDLE SCORE/NUMBER
ODD NUMBERS: 5 10 15
Median = 10
 If N is an EVEN NUMBERS, get the AVERAGE OF TWO MIDDLE
SCORES/NUMBERS
EVEN NUMBERS: 2 4 6 8
Median =
4 + 6
2
Median = 5
DESCRIPTIVE STATISTICS

5.  Mean (for Ungrouped Data): Example
FORMULA:
X =
ΣX
N
X =
(2+4+6+8+10)
5
X =
30
5 X = 6
 Mean (Grouped Data): Example
FORMULA:
X =
ΣfX
N
Class
Interval
118-126
127-135
136-144
145-153
154-162
f
3
5
9
12
5
_______
34
X
122
131
140
149
158
fX
366
655
1260
1788
790
_______
4859
X =
4859
34
X = 142.9
DESCRIPTIVE STATISTICS

6. • Measure of Dispersion
1. Range. Refers simply the highest value minus the lowest value
FORMULA: Range ( R ) = Highest Score (HS) – Lowest Score (LS)
 EXAMPLE:
50 60 10 25 25 30 20 40
R = HS – LS
= 60 – 10
= 50
2 3 4 5 6 7 8 9
R = HS – LS
= 9 – 2
= 7
DESCRIPTIVE STATISTICS

7. 2. Variance. This concerned with the extent to which the values in a data set differ
from the mean.
• Sample Variance. Describes how far the sample scores are spread out around
the sample mean.
FORMULA: SX
2 =
Σ𝑿𝟐 − 𝚺𝑿 𝟐/𝑵
𝑵
Where: SX
2 = sample variance
ΣX = sum of the scores
𝚺𝑿 𝟐 = square sum of the scores
N = number of scores in the sample
DESCRIPTIVE STATISTICS

8. EXAMPLE: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
SX
2 =
𝟔𝟐+ 𝟕𝟐+ 𝟖𝟐+ 𝟗𝟐+ 𝟏𝟎𝟐+ 𝟏𝟏𝟐+ 𝟏𝟐𝟐+ 𝟏𝟑𝟐+ 𝟏𝟒𝟐+ 𝟏𝟓𝟐 − 𝟔+𝟕+𝟖+𝟗+𝟏𝟎+𝟏𝟏+𝟏𝟐+𝟏𝟑+𝟏𝟒+𝟏𝟓 𝟐 /𝟏𝟎
𝟏𝟎
SX
2 =
𝟏, 𝟏𝟖𝟓− 𝟏𝟎𝟓 𝟐 /𝟏𝟎
𝟏𝟎
SX
2 =
𝟏, 𝟏𝟖𝟓−𝟏𝟏,𝟎𝟐𝟓/𝟏𝟎
𝟏𝟎
SX
2 =
𝟏, 𝟏𝟖𝟓−𝟏,𝟏𝟎𝟐.𝟓
𝟏𝟎
SX
2 =
𝟖𝟐.𝟓
𝟏𝟎
SX
2 = 8.25
DESCRIPTIVE STATISTICS

9. • Estimated Population Variance. This is an estimate of how far the scores in
the population would spread out around the population mean.
FORMULA: SX
2 =
Σ𝑿𝟐 − 𝚺𝑿 𝟐/𝑵
𝑵 −𝟏
Where: SX
2 = sample variance
ΣX = sum of the scores
𝚺𝑿 𝟐
= square sum of the scores
N – 1 = number of scores in the sample minus 1
DESCRIPTIVE STATISTICS

10. EXAMPLE: 6, 7, 8, 9, 10, 11, 12, 13, 14, 15
SX
2 =
𝟔𝟐+ 𝟕𝟐+ 𝟖𝟐+ 𝟗𝟐+ 𝟏𝟎𝟐+ 𝟏𝟏𝟐+ 𝟏𝟐𝟐+ 𝟏𝟑𝟐+ 𝟏𝟒𝟐+ 𝟏𝟓𝟐 − 𝟔+𝟕+𝟖+𝟗+𝟏𝟎+𝟏𝟏+𝟏𝟐+𝟏𝟑+𝟏𝟒+𝟏𝟓 𝟐 /𝟏𝟎
𝟏𝟎 −𝟏
SX
2 =
𝟏, 𝟏𝟖𝟓− 𝟏𝟎𝟓 𝟐 /𝟏𝟎
𝟏𝟎 −𝟏
SX
2 =
𝟏, 𝟏𝟖𝟓−𝟏𝟏,𝟎𝟐𝟓/𝟏𝟎
𝟏𝟎 −𝟏
SX
2 =
𝟏, 𝟏𝟖𝟓−𝟏,𝟏𝟎𝟐.𝟓
𝟏𝟎 −𝟏
SX
2 =
𝟖𝟐.𝟓
𝟏𝟎 −𝟏
SX
2 =
𝟖𝟐.𝟓
𝟗
SX
2 =9.17
DESCRIPTIVE STATISTICS

11. 3. Standard Deviation. Refers to the positive square root of the arithmetic mean of
the squared deviations from the mean of the distribution
EQUATION: S2 =
𝑖=1
𝑁
(𝑋1 −X)
2
𝑁
Where: (𝑿 − X)𝟐
= sum of squares of the differences between scores and the mean
N = number of scores in the sample
From the equation, we can simply say that standard deviation is the square root of the
variance, and the variance is the square of the standard deviation
DESCRIPTIVE STATISTICS

12. • Computation of the Standard Deviation for Ungrouped Data
FORMULA: S =
Σ(𝑋−X)2
𝑁
• EXAMPLE
X
15
15
17
18
20
85
𝑋 =
85
5
= 17
(X - 𝑿)
- 2
- 2
- 0
1
3
(𝑋 − X)2
4
4
0
1
9
18
S=
18
5
= 3.6
= 1.9
Step 1:
Compute the
mean
Step 2: Get the deviations from the
mean
Step 3: Square the deviations from
mean
Step 4: Get the sum of the squared deviation
Step 5: Divide
the sum by the
total frequency
Step 5: Extract the square root of the
quotient
DESCRIPTIVE STATISTICS

13. • Computation of the Standard Deviation for Grouped Data
1. Long Method. This method makes use of the following formula:
FORMULA:
s =
Σ𝑓(X2)
N−1
−
(Σ 𝑓𝑥)2
𝑁 (𝑁−1)
2. Coded Deviation Method
FORMULA:
s=
𝑖 Σ 𝑓(𝑑′)2
𝑁 −1
−
(Σ 𝑓𝑑′)2
𝑁 (𝑁 −1)
d′
=
X − X′
𝑖
FORMULA for CODED DEVIATION FROM THE MEAN:
DESCRIPTIVE STATISTICS

14. DESCRIPTIVE STATISTICS
• EXAMPLE: RAW DATA
120
140
161
148
130
156
165
146
142
175
133
150
149
139
143
151
138
150
158
148
180
170
124
161
137
128
147
149
152
142
138
153
168
142
147
118
167
129
130
159

15. 1. Long Method. EXAMPLE
Class
Interval
118-126
127-135
136-144
145-153
154-162
163-171
172-180
f
3
5
9
12
5
4
2
_______
40
X
122
131
140
149
158
167
176
fX
366
655
1260
1788
790
668
352
𝐗 𝟐
14884
17161
19600
22201
24964
27889
30976
𝒇𝐗 𝟐
44652
85805
176400
266412
124820
111556
61952
_______
871597
_______
5879
s =
Σ𝑓(X2)
N−1
−
(Σ 𝑓𝑥)2
𝑁 (𝑁−1)
s =
871597
39
−
58792
40 (39)
s = 22348.64 −
34562641
1560
s = 22348.64 − 22155.54
s = 22348.64 − 22155.54
s = 193.1
s = 13.8
DESCRIPTIVE STATISTICS

16. 2. Coded Deviation Method. EXAMPLE
Class
Interval
118-126
127-135
136-144
145-153
154-162
163-171
172-180
f
3
5
9
12
5
4
2
_______
40
X
122
131
140
149
158
167
176
d’
-3
-2
-1
0
1
2
3
𝒇𝒅′
-9
-10
-9
0
5
8
6
(𝐝′) 𝟐
9
4
1
0
1
4
9
_______
-9
f(𝐝′) 𝟐
27
20
9
0
5
16
18
_______
95
s=
𝑖 Σ 𝑓(𝑑′)2
𝑁 −1
−
(Σ 𝑓𝑑′)2
𝑁 (𝑁 −1)
s=
9 95
39
−
(−9)2
40 (39)
s=
9
2.44 − 0.05
s=
9
2.39
s= 9 (1.55)
s= 13.95
DESCRIPTIVE STATISTICS