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Two clean, exam-ready cheat sheets for Quantitative Methods I: Math and Statistics: including the standard Normal (Z) table and the t-distribution critical values table. 100% free, no signup.

Math Formulas Stats Formulas & Tables

1 · Math Formulas Sheet

Line equations

Point–slope form

y - y_{1} = m (x - x_{1})

Two-point form

y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})

Slope from two points

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Quadratic formula

Roots of ax² + bx + c = 0

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant

Δ = b^{2} - 4 a c

Inverse functions

Power

f (x) = x^{n} \Rightarrow f^{- 1} (x) = x^{1/ n}

Exponential / Logarithm

f (x) = e^{x} \Rightarrow f^{- 1} (x) = ln (x)

General base

f (x) = a^{x} \Rightarrow f^{- 1} (x) = lo g_{a} (x)

Important derivatives

Constant

\frac{d}{d x} c = 0

Power rule

\frac{d}{d x} x^{n} = n x^{n - 1}

Exponential

\frac{d}{d x} e^{x} = e^{x}

General exponential

\frac{d}{d x} a^{x} = a^{x} ln a

Natural log

\frac{d}{d x} ln (x) = \frac{1}{x}

General log

\frac{d}{d x} lo g_{a} (x) = \frac{1}{x l n a}

Derivative rules

Rule	Formula
Sum / Difference	$(f \pm g)^{'} = f^{'} \pm g^{'}$
Constant multiple	$(c f)^{'} = c f^{'}$
Product rule	$(f g)^{'} = f^{'} g + f g^{'}$
Quotient rule	$(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$
Chain rule (composition)	$(f (g (x)))^{'} = f^{'} (g (x)) g^{'} (x)$

Relative rate of increase & elasticity

Relative rate of increase

RRI (x) = \frac{f ^{'} ( x )}{f ( x )} = \frac{d}{d x} ln f (x)

Elasticity

E_{f} (x) = \frac{x}{f ( x )} f^{'} (x) = \frac{d l n f ( x )}{d l n x}

Interpretation: a 1% increase in $x$ changes $f (x)$ by approximately $E_{f} (x)$ %.

Second-order conditions (stationary points)

A stationary point satisfies $f^{'} (x_{0}) = 0$ . Classify with the second derivative:

Condition	Type
$f^{''} (x_{0}) > 0$	Local minimum
$f^{''} (x_{0}) < 0$	Local maximum
$f^{''} (x_{0}) = 0$	Inconclusive: possible saddle / inflection point (check sign change of $f^{'}$ )

For multivariable functions: a critical point is a saddle when the Hessian has eigenvalues of mixed sign ( $f_{xx} f_{y y} - f_{x y}^{2} < 0$ ).

2 · Stats Formulas & Tables Sheet

Descriptive statistics

Range

Range = max - min

IQR

IQR = Q_{3} - Q_{1}

Outlier rule (1.5 × IQR)

x < Q_{1} - 1.5 IQR or x > Q_{3} + 1.5 IQR

Mean

\overset{y}{ˉ} = \frac{1}{n} \sum_{i = 1}^{n} y_{i}

Sample standard deviation

s = \frac{1}{n - 1} \sum_{i = 1}^{n} (y_{i} - \overset{y}{ˉ})^{2}

Z-score

Model-based (population)

z = \frac{y - μ}{σ}

Data-based (sample)

z = \frac{y - y ˉ}{s}

Correlation & linear regression

Correlation coefficient

r = \frac{1}{n - 1} \sum (\frac{x _{i} - x ˉ}{s _{x}}) (\frac{y _{i} - y ˉ}{s _{y}})

Regression line

\overset{y}{^} = b_{0} + b_{1} x

Slope

b_{1} = r \frac{s _{y}}{s _{x}}

Intercept

b_{0} = \overset{y}{ˉ} - b_{1} \overset{x}{ˉ}

Probability rules

Complement

P (A^{c}) = 1 - P (A)

Addition (general)

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Multiplication (general)

P (A \cap B) = P (A) P (B ∣ A)

Conditional

P (B ∣ A) = \frac{P ( A \cap B )}{P ( A )}

Independence

A ⊥ B \Leftrightarrow P (A \cap B) = P (A) P (B)

Random variables

Expected value (discrete)

E (X) = μ = \sum_{x} x P (X = x)

Variance

Var (X) = σ^{2} = \sum_{x} (x - μ)^{2} P (X = x)

Linear transform: mean

E (a X + b) = a E (X) + b

Linear transform: variance

Var (a X + b) = a^{2} Var (X)

Sum of independent X, Y

E (X \pm Y) = E (X) \pm E (Y), Var (X \pm Y) = Var (X) + Var (Y)

Discrete distributions

Distribution	PMF	Mean	SD
Geometric (k = trials until 1st success)	$P (X = k) = (1 - p)^{k - 1} p$	$\frac{1}{p}$	$\frac{1 - p}{p}$
Binomial(n, p)	$(k n) p^{k} (1 - p)^{n - k}$	$n p$	$n p (1 - p)$
Poisson(λ)	$\frac{λ ^{k} e ^{- λ}}{k !}$	$λ$	$λ$

Sampling distribution of ȳ (CLT)

Mean

μ_{\overset{y}{ˉ}} = μ

Standard deviation

σ_{\overset{y}{ˉ}} = \frac{σ}{n}

For large $n$ (rule of thumb $n \geq 30$ ), the sampling distribution of $\overset{y}{ˉ}$ is approximately Normal regardless of the population shape.

Inference framework

Confidence interval

estimate \pm (critical value) \times S E (estimate)

Test statistic

TS = \frac{estimate - null value}{S E ( estimate )}

Standard error (SE) reference table

Parameter	Standard error
Proportion $p$	$S E (\overset{p}{^}) = \frac{p ^ ( 1 - p ^ )}{n}$
Mean $μ$	$S E (\overset{y}{ˉ}) = \frac{s}{n}$
Difference of means $μ_{1} - μ_{2}$	$S E (\overset{y}{ˉ}_{1} - \overset{y}{ˉ}_{2}) = \frac{s _{1}^{2}}{n _{1}} + \frac{s _{2}^{2}}{n _{2}}$
Paired mean $μ_{d}$	$S E (\overset{ˉ}{d}) = \frac{s _{d}}{n}$
Regression slope b₁	$S E (b_{1}) = \frac{s _{e}}{s _{x} n - 1}$
Mean response ŷ at x₀	$S E (\overset{y}{^}) = s_{e} \frac{1}{n} + \frac{( x _{0} - x ˉ ) ^{2}}{( n - 1 ) s _{x}^{2}}$
Prediction at x₀	$S E_{pred} = s_{e} 1 + \frac{1}{n} + \frac{( x _{0} - x ˉ ) ^{2}}{( n - 1 ) s _{x}^{2}}$

Pooled estimates

Pooled proportion

\overset{p}{^}_{pool} = \frac{x _{1} + x _{2}}{n _{1} + n _{2}}

SE: pooled (two proportions)

S E = \overset{p}{^}_{pool} (1 - \overset{p}{^}_{pool}) (\frac{1}{n _{1}} + \frac{1}{n _{2}})

Pooled variance (two means)

s_{p}^{2} = \frac{( n _{1} - 1 ) s _{1}^{2} + ( n _{2} - 1 ) s _{2}^{2}}{n _{1} + n _{2} - 2}

SE: pooled (two means)

S E (\overset{y}{ˉ}_{1} - \overset{y}{ˉ}_{2}) = s_{p} \frac{1}{n _{1}} + \frac{1}{n _{2}}

Chi-square statistic

Chi-square

χ^{2} = \sum \frac{( O - E ) ^{2}}{E}

$O$ = observed count, $E$ = expected count. Degrees of freedom depend on the test (e.g. $(r - 1) (c - 1)$ for an $r \times c$ contingency table).

Table Z: Standard Normal CDF, P(Z ≤ z)

Find row for the integer + 1st decimal of z, column for the 2nd decimal. Two halves shown: negative z (left) and non-negative z (right). Range covered: −3.9 to +3.9.

Negative z values: −3.9 to −0.0
z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
-3.9	0.00005	0.00005	0.00004	0.00004	0.00004	0.00004	0.00004	0.00004	0.00003	0.00003
-3.8	0.00007	0.00007	0.00007	0.00006	0.00006	0.00006	0.00006	0.00005	0.00005	0.00005
-3.7	0.00011	0.00010	0.00010	0.00010	0.00009	0.00009	0.00008	0.00008	0.00008	0.00008
-3.6	0.00016	0.00015	0.00015	0.00014	0.00014	0.00013	0.00013	0.00012	0.00012	0.00011
-3.5	0.00023	0.00022	0.00022	0.00021	0.00020	0.00019	0.00019	0.00018	0.00017	0.00017
-3.4	0.00030	0.00030	0.00030	0.00030	0.00030	0.00030	0.00030	0.00030	0.00030	0.00020
-3.3	0.00050	0.00050	0.00050	0.00040	0.00040	0.00040	0.00040	0.00040	0.00040	0.00030
-3.2	0.00070	0.00070	0.00060	0.00060	0.00060	0.00060	0.00060	0.00050	0.00050	0.00050
-3.1	0.0010	0.00090	0.00090	0.00090	0.00080	0.00080	0.00080	0.00080	0.00070	0.00070
-3.0	0.0013	0.0013	0.0013	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010
-2.9	0.0019	0.0018	0.0018	0.0017	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014
-2.8	0.0026	0.0025	0.0024	0.0023	0.0023	0.0022	0.0021	0.0021	0.0020	0.0019
-2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0026
-2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.0040	0.0039	0.0038	0.0037	0.0036
-2.5	0.0062	0.0060	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
-2.4	0.0082	0.0080	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
-2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
-2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.0110
-2.1	0.0179	0.0174	0.0170	0.0166	0.0162	0.0158	0.0154	0.0150	0.0146	0.0143
-2.0	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
-1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.0250	0.0244	0.0239	0.0233
-1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
-1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
-1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
-1.5	0.0668	0.0655	0.0643	0.0630	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
-1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
-1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
-1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
-1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.1230	0.1210	0.1190	0.1170
-1.0	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
-0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.1660	0.1635	0.1611
-0.8	0.2119	0.2090	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
-0.7	0.2420	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
-0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
-0.5	0.3085	0.3050	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.2810	0.2776
-0.4	0.3446	0.3409	0.3372	0.3336	0.3300	0.3264	0.3228	0.3192	0.3156	0.3121
-0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.3520	0.3483
-0.2	0.4207	0.4168	0.4129	0.4090	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
-0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4364	0.4325	0.4286	0.4247
0.0	0.5000	0.4960	0.4920	0.4880	0.4840	0.4801	0.4761	0.4721	0.4681	0.4641

Non-negative z values: 0.0 to 3.9
z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998
3.6	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	1.0000	1.0000	1.0000
3.9	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

Table T: t-distribution critical values

Two-sided critical values $t^{*}$ for the most common confidence levels (80%, 90%, 95%, 98%, 99%). For a one-sided test at level $α$ , use the column whose two-sided level equals $1 - 2 α$ . The last row ( $\infty$ ) matches z critical values.

Table T: two-sided critical values t* by confidence level and df
df	80%	90%	95%	98%	99%
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169
11	1.363	1.796	2.201	2.718	3.106
12	1.356	1.782	2.179	2.681	3.055
13	1.350	1.771	2.160	2.650	3.012
14	1.345	1.761	2.145	2.624	2.977
15	1.341	1.753	2.131	2.602	2.947
16	1.337	1.746	2.120	2.583	2.921
17	1.333	1.740	2.110	2.567	2.898
18	1.330	1.734	2.101	2.552	2.878
19	1.328	1.729	2.093	2.539	2.861
20	1.325	1.725	2.086	2.528	2.845
21	1.323	1.721	2.080	2.518	2.831
22	1.321	1.717	2.074	2.508	2.819
23	1.319	1.714	2.069	2.500	2.807
24	1.318	1.711	2.064	2.492	2.797
25	1.316	1.708	2.060	2.485	2.787
26	1.315	1.706	2.056	2.479	2.779
27	1.314	1.703	2.052	2.473	2.771
28	1.313	1.701	2.048	2.467	2.763
29	1.311	1.699	2.045	2.462	2.756
30	1.310	1.697	2.042	2.457	2.750
35	1.306	1.690	2.030	2.438	2.724
40	1.303	1.684	2.021	2.423	2.704
45	1.301	1.679	2.014	2.412	2.690
50	1.299	1.676	2.009	2.403	2.678
60	1.296	1.671	2.000	2.390	2.660
80	1.292	1.664	1.990	2.374	2.639
100	1.290	1.660	1.984	2.364	2.626
200	1.286	1.653	1.972	2.345	2.601
500	1.283	1.648	1.965	2.334	2.586
∞ (z)	1.282	1.645	1.960	2.326	2.576

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