Descriptive Statistics Formulas: Sigma, Kurtosis, Skewness, and More
Formulas and Tables
SPC DEMO
Minimize Production Costs, Quickly Detect Issues, and Optimize Your Product Quality
Don’t miss out! Book a demo of our specialized SPC software and unlock immediate improvements in your processes.
- Quality Advisor
- Data Collection Tools
- Data Analysis Tools
- Formulas and Tables
- Glossary
- Additional Resources
Standard normal table
z
x.x0
x.x1
x.x2
x.x3
x.x4
x.x5
x.x6
x.x7
x.x8
x.x9
4.0
.00003
3.9
.00005
.00005
.00004
.00004
.00004
.00004
.00004
.00004
.00003
.00003
3.8
.00007
.00007
.00007
.00006
.00006
.00006
.00006
.00005
.00005
.00005
3.7
.00011
.00010
.00010
.00010
.00009
.00009
.00008
.00008
.00008
.00008
3.6
.00016
.00015
.00015
.00014
.00014
.00013
.00013
.00012
.00012
.00011
3.5
.00023
.00022
.00022
.00021
.00020
.00019
.00019
.00018
.00017
.00017
3.4
.00034
.00032
.00031
.00030
.00029
.00028
.00027
.00026
.00025
.00024
3.3
.00048
.00047
.00045
.00043
.00042
.00040
.00039
.00038
.00036
.00035
3.2
.00069
.00066
.00064
.00062
.00060
.00058
.00056
.00054
.00052
.00050
3.1
.00097
.00094
.00090
.00087
.00084
.00082
.00079
.00076
.00074
.00071
3.0
.00135
.00131
.00126
.00122
.00118
.00114
.00111
.00107
.00104
.00100
2.9
.0019
.0018
.0018
.0017
.0016
.0016
.0015
.0015
.0014
.0014
2.8
.0026
.0025
.0024
.0023
.0023
.0022
.0021
.0021
.0020
.0019
2.7
.0035
.0034
.0033
.0032
.0031
.0030
.0029
.0028
.0027
.0026
2.6
.0047
.0045
.0044
.0043
.0041
.0040
.0039
.0038
.0037
.0036
2.5
.0062
.0060
.0059
.0057
.0055
.0054
.0052
.0051
.0049
.0048
2.4
.0082
.0080
.0078
.0075
.0073
.0071
.0069
.0068
.0066
.0064
2.3
.0107
.0104
.0102
.0099
.0096
.0094
.0091
.0089
.0087
.0084
2.2
.0139
.0136
.0132
.0129
.0125
.0122
.0119
.0116
.0113
.0110
2.1
.0179
.0174
.0170
.0166
.0162
.0158
.0154
.0150
.0146
.0143
2.0
.0228
.0222
.0217
.0212
.0207
.0202
.0197
.0192
.0188
.0183
z
x.x0
x.x1
x.x2
x.x3
x.x4
x.x5
x.x6
x.x7
x.x8
x.x9
1.9
.0287
.0281
.0274
.0268
.0262
.0256
.0250
.0244
.0239
.0233
1.8
.0359
.0351
.0344
.0336
.0329
.0322
.0314
.0307
.0301
.0294
1.7
.0446
.0436
.0427
.0418
.0409
.0401
.0392
.0384
.0375
.0367
1.6
.0548
.0537
.0526
.0516
.0505
.0495
.0485
.0475
.0465
.0455
1.5
.0668
.0655
.0643
.0630
.0618
.0606
.0594
.0582
.0571
.0559
1.4
.0808
.0793
.0778
.0764
.0749
.0735
.0721
.0708
.0694
.0681
1.3
.0968
.0951
.0934
.0918
.0901
.0885
.0869
.0853
.0838
.0823
1.2
.1151
.1131
.1112
.1093
.1075
.1056
.1038
.1020
.1003
.0985
1.1
.1357
.1335
.1314
.1292
.1271
.1251
.1230
.1210
.1190
.1170
1.0
.1587
.1562
.1539
.1515
.1492
.1469
.1446
.1423
.1401
.1379
0.9
.1841
.1814
.1788
.1762
.1736
.1711
.1685
.1660
.1635
.1611
0.8
.2119
.2090
.2061
.2033
.2005
.1977
.1949
.1922
.1894
.1867
0.7
.2420
.2389
.2358
.2327
.2297
.2266
.2236
.2206
.2177
.2148
0.6
.2743
.2709
.2676
.2643
.2611
.2578
.2546
.2514
.2483
.2451
0.5
.3085
.3050
.3015
.2981
.2946
.2912
.2877
.2843
.2810
.2776
0.4
.3446
.3409
.3372
.3336
.3300
.3264
.3228
.3192
.3156
.3121
0.3
.3821
.3783
.3745
.3707
.3669
.3632
.3594
.3557
.3520
.3483
0.2
.4207
.4168
.4129
.4090
.4052
.4013
.3974
.3936
.3897
.3859
0.1
.4602
.4562
.4522
.4483
.4443
.4404
.4364
.4325
.4286
.4247
0.0
.5000
.4960
.4920
.4880
.4840
.4801
.4761
.4721
.4681
.4641
Additional reference material
Additional sections from legacy sigma-f:
Sigma
Sigma of the individuals (actual sigma)
Estimated sigma
Subgroup Size
d2
Subgroup Size
d2
2
1.128
14
3.407
3
1.693
15
3.472
4
2.059
16
3.532
5
2.326
17
3.588
6
2.534
18
3.640
7
2.704
19
3.689
8
2.847
20
3.735
9
2.970
21
3.778
10
3.078
22
3.819
11
3.173
23
3.858
12
3.258
24
3.895
13
3.336
25
3.931
Additional sections from legacy kurtosis-f:
Kurtosis formula
Additional sections from legacy skewness-f:
Skewness formula
Additional sections from legacy coefficient-f:
Coefficient of variance formula
Where
Additional sections from legacy chi-square-f:
Chi-square formula and degrees of freedom table
Alpha value = 5%
Alpha value = 1%
Degrees of freedom
Value
Degrees of freedom
Value
1
3.84
1
6.63
2
5.99
2
9.21
3
7.82
3
11.3
4
9.49
4
13.3
5
11.10
5
15.1
6
12.60
6
16.8
7
14.10
7
18.5
8
15.50
8
20.1
9
16.90
9
23.2
10
18.30
10
24.7
11
19.70
11
26.2
12
21.00
12
27.7
13
22.40
13
29.1
14
23.70
14
30.6
15
25.00
15
30.6
16
26.30
16
32.0
17
27.60
17
33.4
18
28.90
18
34.8
19
30.10
19
36.2
20
31.40
20
37.6
21
32.70
21
38.9
22
33.90
22
40.3
23
35.20
23
41.6
24
36.40
24
43.0
25
37.70
25
44.3
26
38.90
26
45.6
27
40.10
27
47.0
28
41.30
28
48.3
29
42.60
29
49.6
30
43.80
30
50.9
Customers using Advantive in quality advisor
“Enact helps us quickly respond to quality issues, which saves us money.”
““What sets Ben & Jerry’s apart from our competitors is not only our insistence on high-quality ingredients, but also the extra and unique flavours we use to create a euphoric customer experience. Ensuring the final product reflects the passion and quality that we put into each pint required a quality solution that emphasized the same attention to details that we do.””
““By utilizing InfinityQS® ProFicient™ to implement SPC and Six Sigma best practices across our manufacturing processes, Ben & Jerry’s will continue to identify opportunities for cost savings and ensure the highest level of customer satisfaction. The result is the perfect pint for our customers.””