Using Google Sheets statistics functions
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This document provides instructions for using various statistical functions in Google Sheets, including how to construct frequency tables, calculate measures of central tendency and variability, perform hypothesis tests, and analyze correlations and differences between multiple groups. Formulas are presented to calculate probabilities for binomial and normal distributions, simulate random data, estimate sample sizes, and conduct t-tests, ANOVA, and chi-square tests. Videos are linked to demonstrate various statistical analyses and graphs in Google Sheets.
![Simulation with Random Number
• Random between 0 to 1
• =RAND()
• Random within specific range
• =RAND()*([UpperLimit]-[LowerLimit])+[LowerLimit]
• =RANDBETWEEN([LowerLimit],[UpperLimit])
• Normally simulate about 1,000 dataset
• Example
• =let(Price,$B$2,VariableCost,RANDBETWEEN($D$6*100,$E$6*100)/100,DemandQ
ty,RANDBETWEEN($D$7*100,$E$7*100)/100,FixedCost,$B$3,(Price-
VariableCost)*DemandQty-FixedCost)
https://www.indeed.com/career-advice/career-development/how-to-randomize-numbers-in-excel](https://image.slidesharecdn.com/e4-240129091858-b34e0f4b/85/Using-Google-Sheets-statistics-functions-11-320.jpg)
![ANOVA: Two-Factor With Replication
• [A]
• =let(GA,H145:H164,GB,I145:I164,GC,J145:J164,n,count(GA),
sum(sum(GA)^2,sum(GB)^2,sum(GC)^2)/n)
• [BA]
• =let(A,H145:J154,B,H155:J164,n,count(A),sum(sum(A)^2,sum(B)^2)/n)
• [AB]
• =let(GAA,H145:H154,GBA,I145:I154,GCA,J145:J154,GAB,H155:H164,GBB,I155:I164,GCB,J1
55:J164,n,count(GAA),
sum(sum(GAA)^2,sum(GBA)^2,sum(GCA)^2,sum(GAB)^2,sum(GBB)^2,sum(GCB)^2)/n)
• [Y]
• =let(A,H145:J154,B,H155:J164,SUMSQ(A,B))
• [T]
• =let(A,H145:J154,B,H155:J164,SUM(A,B)^2/count(A,B))](https://image.slidesharecdn.com/e4-240129091858-b34e0f4b/85/Using-Google-Sheets-statistics-functions-24-320.jpg)
Using Google Sheets statistics functions
- 1. a111chen @2023 Using Google Sheets For Statistics
- 2. Objective • Try to use Google Sheets function especially : LET, ARRAYFORMULA • For function LET, • It can show how the variable in the formula, the variable can be in array or range • We can compare with different statistic formula and more easy. • In future, just copy the formula, and change the variable, and boom finished. • For function ARRAYFORMULA, • In my opinion, Google Sheets Arrayformula is more easy to detect mistake at the formula compare with Mircosoft Excel
- 3. Table of Content • Construct Frequency Table with Normal Distribution • Normality testing • Normal Approximate to Binomial Distribution • Sample Size • Hypothesis Testing • One Sample • Two Sample • Correlation between Two Sample • K-Independent Sample
- 4. Construct Frequency Table • Lowest • =MIN(Dataset) • Highest • =MAX(Dataset) • Total Class No • =ROUNDUP(LN(COUNT(Dataset))/LN(2),0) • Class Width • =ROUNDUP((Highest-Lowest)/TotalNo,0) • Frequency • =COUNTIFS(Dataset,">="&Lower,Dataset,"<"&Upper) https://www.youtube.com/watch?v=YfVu7xGHgnA
- 5. Frequency Graph 0 5 10 15 20 25 70 - 75 75 - 80 80 - 85 85 - 90 90 - 95 95 - 100 Class Boundaries Frequency (Min) (Max) (Mean) (Median) (Q1) (Q3) https://www.youtube.com/watch?v=39lsUsJsc2c
- 6. What graph can show the Normality? 8.3% 5.0% 20.0% 21.7% 38.3% 6.7% 70 - 75 75 - 80 80 - 85 85 - 90 90 - 95 95 - 100 Class Boundaries Skew: -0.75, Kurt:0.06 Frequency in % with Normal Distribution Count in % -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 -3 -2 -1 0 1 2 3 Data quantiles (Z-score) Normal theoretical quantile (Z-score) QQ Plot https://www.youtube.com/watch?v=g5DTW2IQwxk
- 7. Construct Normal Distribution • Normal Distribution • =NORM.DIST(Midpoint,Mean,SD,FALSE)*ClassWidth • Normal theoretical quantile (Z-score) • =NORM.S.INV((RANK.AVG(x,Dataset,1)-0.5)/n) • Data quantiles (Z-score) • =STANDARDIZE(x,Mean,SD) • Normality test • Skew • =SKEW(Dataset) • Kurt • =KURT(Dataset) • Positive skewness extending toward more positive values. • Negative skewness extending toward more negative values. • Positive kurtosis indicates a relatively peaked distribution. • Negative kurtosis indicates a relatively flat distribution.
- 8. Common use • Total sample number, n • =COUNT(Dataset) • =let(f,F2:F6, SUM(f)) • Mean • =AVERAGE(Dataset) • =let(n,F7,x,$E$2:$E$6,f,F2:F6, SUMPRODUCT(x,f)/n) • Standard deviation,SD • =STDEV.S(Dataset) • =let(n,F7, x,$E$2:$E$6,f,F2:F6, Mean,F8, sqrt(SUMPRODUCT(f,x^2)/n-Mean^2))
- 9. Binomial Distribution • Number of trials, n • probability of success, p • P(X=x) • = BINOM.DIST(x,n,p,FALSE) • Check Probability between (from table) • =PROB(RangeX,RangeP(X=x),LowerLimit,UpperLimit) • =SUMIFS(RangeP(X=x), RangeX,">="& LowerLimit, RangeX,"<="& UpperLimit) PROB returns the error value. • If any value in prob_range ≤ 0 or if any value in prob_range > 1, • If the sum of the values in prob_range is not equal to 1, • If x_range and prob_range contain a different number of data points,
- 10. Normal Approximate to Binomial Distribution • Rough guideline: • np >=10 and n(1-p) >=10 • Example • n = 75, p = 0.6 • Mean = 45 • Std Dev. = 4.24 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 n = 75, p = 0.6, Mean = 45, Std Dev. = 4.24 Normal Approximate to Binomial Distribution P(X=x) x Norm https://www.youtube.com/watch?v=CCqWkJ_pqNU&t=313s https://www.youtube.com/watch?v=A2sd09qCvcg
- 11. Simulation with Random Number • Random between 0 to 1 • =RAND() • Random within specific range • =RAND()*([UpperLimit]-[LowerLimit])+[LowerLimit] • =RANDBETWEEN([LowerLimit],[UpperLimit]) • Normally simulate about 1,000 dataset • Example • =let(Price,$B$2,VariableCost,RANDBETWEEN($D$6*100,$E$6*100)/100,DemandQ ty,RANDBETWEEN($D$7*100,$E$7*100)/100,FixedCost,$B$3,(Price- VariableCost)*DemandQty-FixedCost) https://www.indeed.com/career-advice/career-development/how-to-randomize-numbers-in-excel
- 13. Hypothesis Testing Type I - Error • critical value of the sample mean (1) • =let(n,B4,mean,B2,SD,B3,Alpha,B7, mean+ CONFIDENCE.NORM(Alpha,SD,n)) • critical value of the sample mean (2) • =let(n,B4,mean,B2,SD,B3,Alpha,B7, mean- CONFIDENCE.NORM(Alpha,SD,n)) Type II - Error • Power of the test • =let(n,B4, mean,B12, SD,B3, cvalue,B13, SE,SD/SQRT(n),Zscore,(cvalue- mean)/SE,Beta,NORM.S.DIST(Zscore)-NORM.S.DIST(0)+0.5,1-Beta) • Reject (Power of test <0.8) or accept
- 14. T-test for One sample, and Parametric • T-test • =let(n,B25,Meanp,B26,Means,B27,SDs,B28, SE,SDs/SQRT(n),(Means-Meanp)/SE) • T-test critical • =let(n,B25, alpha,B24,DF,n-1,abs(T.INV(alpha,DF))) • Reject (t > t-test critical) or accept Note: • Parametric Data is Normal distributed • Non Parametric Data is not Normal distributed
- 15. Observe Data and Expected Data, What test? • Independent • Chi-Square : ⅀ (Oi - Ei)^2/Ei • =let(OA,B82:B84,OB,C82:C84,EA,E82:E84, EB,F82:F84, sum(ARRAYFORMULA((OA- EA)^2/EA),ARRAYFORMULA((OB- EB)^2/EB))) • Chi-Square Critical • =let(nCol,B74,nRow,B75, alpha,B77,DF,(nCol-1)*(nRow- 1),CHISQ.INV.RT(alpha,DF)) • Reject (Chi-Square > Chi-square critical ) or accept • P-Value • =let(nCol,B74,nRow,B75, chisquare,B88, df,(nCol-1)*(nRow-1), CHISQ.DIST.RT(chisquare,df)) • =Let(Oi,B82:C84,Ei,E82:F84, CHISQ.TEST(Oi,Ei)) 0 5 10 15 20 25 30 35 40 Yes (O1) No (O2) Yes (E1) No (E2) Observe Expected Heavy smoker Moderate Nonsmoker Eij = Sum of Col(i)* Sum of Row(j)/(Total sum of Col or Row)
- 16. Chi-square Test Related • McNemar test • =let(FF,B130,NFNF,C131,(abs(FF- NFNF)-1)^2/(FF+NFNF)) • Chi-square critical • =let(nCol,B122,nRow,B123, alpha,B124,DF,(nCol-1)*(nRow- 1),CHISQ.INV.RT(alpha,DF)) • Reject (McNemar > Chi-square critical ) or accept 10 60 90 40 Favor Not Favor Before/ After Favor Not Favor
- 17. Chi-Square related Statistic • Statistic, Phi • =let(ChiSquare,B62,N,B60, sqrt((Chisquare/N))) • Statistic, Cramer’s V • =let(ChiSquare,B62,N,B60,K,B61,sqrt((ChiSquare/(N*(K-1))))) • Statistic, Contigency coefficienct C • =let(ChiSquare,B62,N,B60, sqrt((ChiSquare/(ChiSquare+N)))) • Probability of error, P(A) • =let(n,D74,totalCol,B74:C74,PercentCol,B75:C75, 1- SUMPRODUCT(totalCol,PercentCol)/n) • Probability of error, P(B) • =let(OA,B71:B73,OB,C71:C73,TotalRow,D71:D73,n,D74,1- sum(ARRAYFORMULA(OA^2/TotalRow+OB^2/TotalRow))/n) • Goodman & Kruskal tau: • =Let(PA,B81,PB,B82,(PA-PB)/PA)
- 18. Correlation Graph, How to test? -1000 0 1000 2000 3000 4000 5000 6000 -5 0 5 10 15 20 25 30 Y (Price) X (Temperature)
- 19. T-Test Correlation Test • Pearson correlation (Parametric) • =let(n,F4,y,B3:B12,x,C3:C12,r,PEARSON(y,x), r/sqrt((1-r^2)/(n-2))) • =let(n,F4,y,B3:B12,x,C3:C12,r, CORREL(y,x), r/sqrt((1-r^2)/(n-2))) • =let(y,B3:B12,x,C3:C12,n,count(x),a,n*SUMPRODUCT(x,y)- (sum(x)*sum(y)),b,n*SUMPRODUCT(x^2)-sum(x)^2,c,n*SUMPRODUCT(y^2)- sum(y)^2,a/SQRT(b*c)) • Spearman’s Rho (Non Parametric) • =let(x,E114:E123,y,F114:F123,n,count(x),RankX,MAP(x,LAMBDA(r,RANK.AVG(r,x))),Ran kY,MAP(y,LAMBDA(r,RANK.AVG(r,y))),rs,1-6*SUMXMY2(RankX,RankY)/(n^3- n),rs*sqrt((n-2)/(1-rs^2))) • T-Test Critical • =let(n,F4, alpha,F7,df,n-2,abs(T.INV(alpha,df))) • Reject (Pearson correlation > T-Test critical) or accept • Reject (Spearman’s Rho > T-Test critical) or accept
- 20. F-Test Correlation Test • F-Ratio (Parametric) • =let(n,B41,k,B42,x,C23:C32,y,B23:B32,Intercept,INTERCEPT(y,x),Slope,SLOPE(y,x), RegMeanSq,devsq(ARRAYFORMULA(x*Slope+Intercept))/(k-1), ResidualMeanSq, SUMXMY2(ARRAYFORMULA(x*Slope+Intercept),y)/(n-k), RegMeanSq/ResidualMeanSq) • F-test Critical • =let(n,B41,k,B42,dfA,k-1,dfB,n-k,F.INV.RT(0.05,dfA,dfB)) • Reject (F-Ratio > F-Test critical) or accept
- 21. T-Test Two Sample Test, Parametric • Related • T-test • =let(YA,B103:B112,YB,C103:C112,n,B97,df,n-1,sd,sqrt((SUMXMY2(YA,YB)- sum(ARRAYFORMULA(YA-YB))^2/n)/df),(sum(ARRAYFORMULA(YA- YB))/n)/(sd/sqrt(n))) • T-test critical • =let(n,B97, alpha,B98,df,n-1, abs(T.INV.2T(alpha,df))) • Reject (T-Test > T-Test critical) or accept • P-Value • =let(n,B97,df,n-1,Ttest,B115,T.DIST.2T(Ttest,df)) • =let(YA,B103:B112,YB,C103:C112,T.TEST(YA,YB,2,1)) tails - Specifies the number of distribution tails. • If 1: uses a one-tailed distribution. • If 2: uses a two-tailed distribution. type - Specifies the type of t-Test. • If 1: a paired test is performed. • If 2: a two-sample equal variance (homoscedastic) test is performed. • If 3: a two-sample unequal variance (heteroscedastic) test is performed. T.INV(alpha,df) = T.INV.2T(2*alpha,df)
- 22. K-Independent Sample 0 10 20 30 40 50 60 70 80 90 100 1 2 3 k (number of groups) 1 2 b (number of replicate) n (Total number of group interaction)
- 23. Analysis of variance (One Way) • F Value • =let(K,B140,GA,H145:H164,GB,I145:I164,GC,J145:J164,N,count(GA,GB,GC), nG,count(GA),sum2n,sum(GA,GB,GC)^2/N, sumx2n,sum(sum(GA)^2,sum(GB)^2,sum(GC)^2)/nG,sumSQn,sum(SUMSQ(GA),SU MSQ(GB),SUMSQ(GC)),SSBetween,(sumx2n-sum2n)/(k-1),SSWithin,(sumSQn- sumx2n)/(n-k),SSBetween/SSWithin) • F-test Critical value • =let(n,B139,k,B140,dfA,k-1,dfB,n-k,F.INV.RT(0.05,dfA,dfB))
- 24. ANOVA: Two-Factor With Replication • [A] • =let(GA,H145:H164,GB,I145:I164,GC,J145:J164,n,count(GA), sum(sum(GA)^2,sum(GB)^2,sum(GC)^2)/n) • [BA] • =let(A,H145:J154,B,H155:J164,n,count(A),sum(sum(A)^2,sum(B)^2)/n) • [AB] • =let(GAA,H145:H154,GBA,I145:I154,GCA,J145:J154,GAB,H155:H164,GBB,I155:I164,GCB,J1 55:J164,n,count(GAA), sum(sum(GAA)^2,sum(GBA)^2,sum(GCA)^2,sum(GAB)^2,sum(GBB)^2,sum(GCB)^2)/n) • [Y] • =let(A,H145:J154,B,H155:J164,SUMSQ(A,B)) • [T] • =let(A,H145:J154,B,H155:J164,SUM(A,B)^2/count(A,B))
- 25. ANOVA: Two-Factor With Replication • Within Group (S/AB) • =Let(Y,E178,AB,E177,k,B168,b,B169,n,B170,(Y-AB)/(k*b*(n-1))) • F Value - Between Group A • =Let(Y,E178,AB,E177,k,B168,b,B169,n,B170,A,E175,T,E179,Within,D187,(A-T)/(k- 1)/Within) • F Value - Between Group B • =Let(Y,E178,AB,E177,k,B168,b,B169,n,B170,BA,E176,T,E179,Within,D187,(BA-T)/(b- 1)/Within) • F Value - Interaction (AxB) • =Let(Y,E178,AB,E177,k,B168,b,B169,n,B170,BA,E176,A,E175,T,E179,Within,D187,(AB- A-BA+T)/(k-1)/(b-1)/Within) • F-test Critical value • =let(k,$B$168,b,$B$169,n,$B$170, dfWithin,(k*b*(n- 1)),df,C184,F.INV.RT(0.05,df,dfWithin)) • Reject (F Value > F-Test critical) or accept
- 26. Conclusion • Already show some example of function LET, ARRAYFORMULA in statistics • If need more further understand the statistic for business use, maybe can refer to • Main Reference: Business Research Methods, Pamela Schindler, 14th Edition. • Beside from the understand above example, it also can • Applied to business use, for example marketing research, organization behavior research • Combine with lambda function, for our example Spearman’s Rho • Create new formula