Standard deviation error bars for each interpolated value of temperature

I am trying to produce standard deviation error bars for the interpolated data ranging from 10 to 35 degrees with a a 1 degree interval, being plotted on the averaged interpolated data line. However, I can't seem to produce seperate standard deviations for each of the interpolated temperatures.

Below is the code I have used for one of the 4 oils.

Oilviscositytest10cst = readmatrix('Oil viscosity test 10cst.xlsx','Range','A6:O65');
xq = linspace(10,35,25);
% 10 cSt oil
% run 1
figure(1)
% run 1 ramp up
temp_1_rup_10 = Oilviscositytest10cst(2:27,3);
viscosity_1_rup_10 = Oilviscositytest10cst(2:27,5);
vq_1_up_10 = interp1(temp_1_rup_10,viscosity_1_rup_10,xq);
plot(xq,vq_1_up_10,'r')
hold on
plot(temp_1_rup_10,viscosity_1_rup_10,'ko')
title('10 cSt oil')
% run 1 ramp down
temp_1_rdown_10 = Oilviscositytest10cst(35:60,3);
viscosity_1_rdown_10 = Oilviscositytest10cst(35:60,5);
vq_1_down_10 = interp1(temp_1_rdown_10,viscosity_1_rdown_10,xq);
plot(xq,vq_1_down_10,'b')
hold on
plot(temp_1_rdown_10,viscosity_1_rdown_10,'ko')
% run 2
% run 2 ramp up
temp_2_rup_10 = Oilviscositytest10cst(2:27,11);
viscosity_2_rup_10 = Oilviscositytest10cst(2:27,13);
vq_2_up_10 = interp1(temp_2_rup_10,viscosity_2_rup_10,xq);
plot(xq,vq_2_up_10,'c')
hold on
plot(temp_2_rup_10,viscosity_2_rup_10,'ko')
% run 2 ramp down
temp_2_rdown_10 = Oilviscositytest10cst(35:60,11);
viscosity_2_rdown_10 = Oilviscositytest10cst(35:60,13);
vq_2_down_10 = interp1(temp_2_rdown_10,viscosity_2_rdown_10,xq);
plot(xq,vq_2_down_10,'g')
hold on
plot(temp_2_rdown_10,viscosity_2_rdown_10,'ko')
% run 3
% run 3 ramp up
temp_3_rup_10 = Oilviscositytest10cst(2:27,3);
viscosity_3_rup_10 = Oilviscositytest10cst(2:27,5);
vq_3_up_10 = interp1(temp_3_rup_10,viscosity_3_rup_10,xq);
plot(xq,vq_3_up_10,'r')
hold on
plot(temp_3_rup_10,viscosity_3_rup_10,'ko')
% run 3 ramp down
temp_3_rdown_10 = Oilviscositytest10cst(35:60,3);
viscosity_3_rdown_10 = Oilviscositytest10cst(35:60,5);
vq_3_down_10 = interp1(temp_3_rdown_10,viscosity_3_rdown_10,xq);
plot(xq,vq_3_down_10,'b')
hold on
plot(temp_3_rdown_10,viscosity_3_rdown_10,'ko')
% average viscosity
six = 6*(ones(1,1));
addition_vq_10 = vq_1_up_10 + vq_1_down_10 + vq_2_up_10 + vq_2_down_10 + vq_3_up_10 + vq_3_down_10;
average_vq_10 = addition_vq_10/six;
plot(xq,average_vq_10,'k')
plot(xq, average_vq_10, 'ko')
err_10 = (std(average_vq_10,[],2)/sqrt(size(average_vq_10,2)))

err_10 = 0.2498

e_10 = errorbar(xq,average_vq_10,err_10);

figure(2)
plot(xq,average_vq_10,'m')
hold on 
plot(xq, average_vq_10, 'ko')

Answer 1

[Torsten]

You mean the graphics if you add the line

errorbar(xq,average_vq_10,err_10,"blue")

at the end of your code from above ?

[Daniel Jackson]

The issue with the code at the end is that it produces the same standard deviation for all of the temperatures. I have changed it to this code, and it seems to produce something more expected.

a_10 = (vq_1_down_10 - average_vq_10).^2;

b_10 = (vq_1_up_10 - average_vq_10).^2;

c_10 = (vq_2_down_10 - average_vq_10).^2;

d_10 = (vq_2_up_10 - average_vq_10).^2;

e_10 = (vq_3_down_10 - average_vq_10).^2;

f_10 = (vq_3_up_10 - average_vq_10).^2;

sum_10 = (a_10 + b_10 + c_10 + d_10 + e_10 + f_10);

division_10 = sum_10/6;

S_10 = sqrt(division_10);

e_10 = errorbar(xq,average_vq_10,S_10);

[Torsten]

I wonder how you compute standard deviations for interpolated values. Doesn't the error depend on the interpolation method you use ?

[Daniel Jackson]

Well it's 6 seperate tests which have all been interpolated to then produce an average. So i believe it's the standard deviation of them 6 tests, where I would then plot the standard deviation error bar onto the averaged data line, thus allowing me to analyse the data and state how repeatable said experiment is. If you get what I'm trying to say.

[Daniel Jackson]

The standard deviation is calculated for the 6 data points (3 from the up-ramp and 3 from the down-ramp) for each temperature (after interpolation). The standard deviation should be different for each temperature point.

[Torsten]

Why do you use the data from run 1 in run 3 ?

Shouldn't the columns be 19 and 21 instead of 3 and 5 ?

And I'd compute the errorbar length in each point xq from the six values of the six measurements, i.e.:

meanvalue(i) = addition_vq_10(i)/6;
err_10(i) = sqrt(((vq_1_down_10(i)-meanvalue(i))^2 +(vq_1_up_10(i)-meanvalue(i))^2+...+(vq_3_up_10(i)-meanvalue(i))^2)/5)

[Daniel Jackson]

They should be yes, didn't spot that. Thank you

[Daniel Jackson]

Would you then add them all together and then plot??

[Daniel Jackson]

I've tried that code and the error message "Unrecognized function or variable 'i'." occurs

[Torsten]

It seems we computed the same.

But still I'm not convinced whether it's statistically correct to interpolate the six experiments to equal values of temperature and then compute errorbars for the interpolated values.

Oilviscositytest10cst = readmatrix('Oil viscosity test 10cst.xlsx','Range','A6:W65');
xq = linspace(10,35,25);
% 10 cSt oil
% run 1
figure(1)
% run 1 ramp up
temp_1_rup_10 = Oilviscositytest10cst(2:27,3);
viscosity_1_rup_10 = Oilviscositytest10cst(2:27,5);
vq_1_up_10 = interp1(temp_1_rup_10,viscosity_1_rup_10,xq);
plot(xq,vq_1_up_10,'r')
hold on
plot(temp_1_rup_10,viscosity_1_rup_10,'ko')
title('10 cSt oil')
% run 1 ramp down
temp_1_rdown_10 = Oilviscositytest10cst(35:60,3);
viscosity_1_rdown_10 = Oilviscositytest10cst(35:60,5);
vq_1_down_10 = interp1(temp_1_rdown_10,viscosity_1_rdown_10,xq);
plot(xq,vq_1_down_10,'b')
hold on
plot(temp_1_rdown_10,viscosity_1_rdown_10,'ko')
% run 2
% run 2 ramp up
temp_2_rup_10 = Oilviscositytest10cst(2:27,11);
viscosity_2_rup_10 = Oilviscositytest10cst(2:27,13);
vq_2_up_10 = interp1(temp_2_rup_10,viscosity_2_rup_10,xq);
plot(xq,vq_2_up_10,'c')
hold on
plot(temp_2_rup_10,viscosity_2_rup_10,'ko')
% run 2 ramp down
temp_2_rdown_10 = Oilviscositytest10cst(35:60,11);
viscosity_2_rdown_10 = Oilviscositytest10cst(35:60,13);
vq_2_down_10 = interp1(temp_2_rdown_10,viscosity_2_rdown_10,xq);
plot(xq,vq_2_down_10,'g')
hold on
plot(temp_2_rdown_10,viscosity_2_rdown_10,'ko')
% run 3
% run 3 ramp up
temp_3_rup_10 = Oilviscositytest10cst(2:27,19);
viscosity_3_rup_10 = Oilviscositytest10cst(2:27,21);
vq_3_up_10 = interp1(temp_3_rup_10,viscosity_3_rup_10,xq);
plot(xq,vq_3_up_10,'r')
hold on
plot(temp_3_rup_10,viscosity_3_rup_10,'ko')
% run 3 ramp down
temp_3_rdown_10 = Oilviscositytest10cst(35:60,19);
viscosity_3_rdown_10 = Oilviscositytest10cst(35:60,21);
vq_3_down_10 = interp1(temp_3_rdown_10,viscosity_3_rdown_10,xq);
plot(xq,vq_3_down_10,'b')
hold on
plot(temp_3_rdown_10,viscosity_3_rdown_10,'ko')

% average viscosity
addition_vq_10 = vq_1_up_10 + vq_1_down_10 + vq_2_up_10 + vq_2_down_10 + vq_3_up_10 + vq_3_down_10;
average_vq_10 = addition_vq_10/6;
err_10 = sqrt(((vq_1_down_10-average_vq_10).^2 +...
               (vq_1_up_10-average_vq_10).^2+...
               (vq_2_down_10-average_vq_10).^2 +...
               (vq_2_up_10-average_vq_10).^2+...
               (vq_3_down_10-average_vq_10).^2+...
               (vq_3_up_10-average_vq_10).^2)/5);
figure(2)
plot(xq,average_vq_10,'m')
hold on 
plot(xq, average_vq_10, 'ko')
errorbar(xq,average_vq_10,err_10);

Answer 2

[Star Strider]

It is difficult to follow your code, so I created my own based on my best guess as to what you want.

Try this —

% readcell('Oil viscosity test 10cst.xlsx')
% opts = spreadsheetImportOptions('VariableNamesRange','A2:W2', 'VariableUnitsRange','B6:W6', 'VariableNamingRule','preserve')
T1 = readtable('Oil viscosity test 10cst.xlsx', 'VariableNamingRule','preserve')

T1 = 60×23 table
    Point No.    Time     Temperature    Shear Stress    Viscosity    Torque       Status       Var8    Point No._1    Time_1    Temperature_1    Shear Stress_1    Viscosity_1    Torque_1     Status_1      Var16    Point No._2    Time_2    Temperature_2    Shear Stress_2    Viscosity_2    Torque_2     Status_2  
    _________    _____    ___________    ____________    _________    _______    ___________    ____    ___________    ______    _____________    ______________    ___________    ________    ___________    _____    ___________    ______    _____________    ______________    ___________    ________    ___________

       NaN         NaN         NaN             NaN           NaN          NaN    {0×0 char }    NaN         NaN          NaN           NaN               NaN             NaN           NaN     {0×0 char }     NaN         NaN          NaN           NaN               NaN             NaN            NaN    {0×0 char }
         1       0.333          10          2.2877        11.441      0.12939    {'Dy_auto'}    NaN           1        0.333           9.9            2.3499          11.752       0.13291     {'Dy_auto'}     NaN           1        0.333           9.9           0.58291          11.661       0.032968    {'Dy_auto'}
         2       0.667       10.78          2.2549        11.275      0.12753    {'Dy_auto'}    NaN           2        0.667         10.71            2.3266          11.634       0.13159     {'Dy_auto'}     NaN           2        0.667         10.72             0.575          11.501       0.032521    {'Dy_auto'}
         3           1       11.79          2.2244        11.122      0.12581    {'Dy_auto'}    NaN           3            1         11.73            2.2859           11.43       0.12929     {'Dy_auto'}     NaN           3            1         11.74           0.56589          11.318       0.032005    {'Dy_auto'}
         4       1.333       12.85          2.1774        10.887      0.12315    {'Dy_auto'}    NaN           4        1.333         12.81            2.2371          11.186       0.12653     {'Dy_auto'}     NaN           4        1.333         12.82            0.5541          11.082       0.031339    {'Dy_auto'}
         5       1.667       13.91          2.1301         10.65      0.12047    {'Dy_auto'}    NaN           5        1.667          13.9            2.1851          10.925       0.12358     {'Dy_auto'}     NaN           5        1.667         13.89           0.54295          10.859       0.030708    {'Dy_auto'}
         6           2       14.97          2.0879        10.439      0.11808    {'Dy_auto'}    NaN           6            2         14.96            2.1448          10.724        0.1213     {'Dy_auto'}     NaN           6            2         14.95           0.53163          10.633       0.030068    {'Dy_auto'}
         7       2.333       16.01          2.0485        10.242      0.11586    {'Dy_auto'}    NaN           7        2.333         16.01            2.0986          10.493       0.11869     {'Dy_auto'}     NaN           7        2.333            16            0.5213          10.426       0.029484    {'Dy_auto'}
         8       2.667       17.04          2.0123        10.061      0.11381    {'Dy_auto'}    NaN           8        2.667         17.05            2.0534          10.267       0.11614     {'Dy_auto'}     NaN           8        2.667         17.03           0.50957          10.191        0.02882    {'Dy_auto'}
         9           3       18.06          1.9715        9.8573       0.1115    {'Dy_auto'}    NaN           9            3         18.07            2.0176          10.088       0.11411     {'Dy_auto'}     NaN           9            3         18.06           0.49942          9.9885       0.028246    {'Dy_auto'}
        10       3.333       19.08          1.9365        9.6824      0.10952    {'Dy_auto'}    NaN          10        3.333         19.09            1.9807          9.9033       0.11202     {'Dy_auto'}     NaN          10        3.333         19.08           0.49135           9.827        0.02779    {'Dy_auto'}
        11       3.667       20.09          1.9012        9.5058      0.10753    {'Dy_auto'}    NaN          11        3.667          20.1            1.9439          9.7196       0.10994     {'Dy_auto'}     NaN          11        3.667         20.09           0.48154          9.6309       0.027235    {'Dy_auto'}
        12           4       21.12          1.8679        9.3396      0.10564    {'Dy_auto'}    NaN          12            4         21.11            1.9082          9.5409       0.10792     {'Dy_auto'}     NaN          12            4         21.12           0.47284          9.4568       0.026743    {'Dy_auto'}
        13       4.333       22.13          1.8342        9.1709      0.10374    {'Dy_auto'}    NaN          13        4.333         22.12            1.8721          9.3607       0.10588     {'Dy_auto'}     NaN          13        4.333         22.13           0.46581          9.3162       0.026345    {'Dy_auto'}
        14       4.667       23.14          1.8071        9.0355      0.10221    {'Dy_auto'}    NaN          14        4.667         23.11            1.8464          9.2318       0.10443     {'Dy_auto'}     NaN          14        4.667         23.13           0.45763          9.1526       0.025882    {'Dy_auto'}
        15           5       24.14          1.7729        8.8647      0.10027    {'Dy_auto'}    NaN          15            5         24.13             1.811          9.0549       0.10242     {'Dy_auto'}     NaN          15            5         24.13           0.44973          8.9946       0.025436    {'Dy_auto'}

Mv = all(ismissing(T1(:,1:5)),2);
T1(Mv,:) = [];

ColIdx = [1 3 5] + [0; 8; 16]

ColIdx = 3×3

1 3 5 9 11 13 17 19 21

xq = linspace(10,35,25);                                                                            % Temperature Vector For Interpolation

for  k = 1:size(ColIdx,1)
    % k
    idx = cumsum([true; diff(T1{:,ColIdx(k,1)})<0]);
    TempVisc = accumarray(idx, (1:numel(idx)).', [], @(x){[T1{x,ColIdx(k,2)} T1{x,ColIdx(k,3)}]});  % Accumulate Results
    % TempVisc
    Temp{k} = [TempVisc{1}(:,1) flipud(TempVisc{2}(:,1))];
    Visc{k} = [TempVisc{1}(:,2) flipud(TempVisc{2}(:,2))];                                                         % unique Values For Interpolation
    Viscic{k,1} = interp1(Temp{k}(:,1), Visc{k}(:,1), xq(:));                                          % Interpolate
    Viscic{k,2} = interp1(Temp{k}(:,2), Visc{k}(:,2), xq(:)); 
end

Temp

Temp = 1×3 cell array
    {26×2 double}    {26×2 double}    {26×2 double}

Visc

Visc = 1×3 cell array
    {26×2 double}    {26×2 double}    {26×2 double}

Viscic

Viscic = 3×2 cell array
    {25×1 double}    {25×1 double}
    {25×1 double}    {25×1 double}
    {25×1 double}    {25×1 double}

Visci = cell2mat(reshape(Viscic, 1, []))

Visci = 25×6

11.4410 11.7374 11.6415 11.6115 11.5984 11.4241 11.2354 11.5677 11.4433 11.4035 11.3656 11.2047 11.0570 11.3502 11.2430 11.1709 11.1697 10.9778 10.8255 11.1106 11.0184 10.9674 10.9653 10.7750 10.5989 10.8744 10.8000 10.7581 10.7561 10.5899 10.3939 10.6694 10.5821 10.5267 10.5238 10.3743 10.1998 10.4408 10.3690 10.3228 10.3197 10.1738 10.0107 10.2246 10.1396 10.1478 10.1441 9.9915 9.8104 10.0403 9.9452 9.9728 9.9700 9.8093 9.6308 9.8515 9.7697 9.7777 9.7759 9.6285

Tempm = cell2mat(Temp)

Tempm = 26×6

10.0000 9.9000 9.9000 9.9100 9.9000 9.9000 10.7800 10.9000 10.7100 10.9000 10.7200 10.9000 11.7900 11.8900 11.7300 11.9000 11.7400 11.8900 12.8500 12.8900 12.8100 12.9000 12.8200 12.8900 13.9100 13.8800 13.9000 13.8900 13.8900 13.8900 14.9700 14.8800 14.9600 14.8900 14.9500 14.8900 16.0100 15.8800 16.0100 15.9000 16.0000 15.8800 17.0400 16.8800 17.0500 16.9000 17.0300 16.8800 18.0600 17.9000 18.0700 17.9000 18.0600 17.8800 19.0800 18.8800 19.0900 18.9000 19.0800 18.8900

Viscm = cell2mat(Visc)

Viscm = 26×6

11.4410 11.6310 11.7520 11.6190 11.6610 11.4450 11.2750 11.4360 11.6340 11.3920 11.5010 11.2360 11.1220 11.2090 11.4300 11.2060 11.3180 11.0170 10.8870 11.0120 11.1860 11.0080 11.0820 10.8140 10.6500 10.8240 10.9250 10.8200 10.8590 10.6480 10.4390 10.5940 10.7240 10.5890 10.6330 10.4380 10.2420 10.3890 10.4930 10.3820 10.4260 10.2400 10.0610 10.2100 10.2670 10.2040 10.1910 10.0610 9.8573 10.0560 10.0880 10.0510 9.9885 9.8922 9.6824 9.8678 9.9033 9.8640 9.8270 9.7075

figure
plot(Tempm, Viscm)
grid
xlabel('Temperature (°C)')
ylabel('Viscosity (mPa\cdots)')
title('Viscosity vs. Temperature')
axis('padded')

ViscMean = mean(Visci, 2);                                                                         % Mean Of Interpolated Viscosity Vectors
ViscSE = std(Visci, [], 2) / sqrt(size(Visci,2));                                                  % Standared Error Of Interpolated Viscosity Vectors

figure
errorbar(xq, ViscMean, ViscSE)
grid
xlabel('Temperature (°C)')
ylabel('Viscosity (mPa\cdots)')
title('Mean Viscosity vs. Temperature With Errors')
axis('padded')

return

Oilviscositytest10cst = table2array(T1(:, [1:6 8:14 16:end-1]))
T1(25:28,:)
xq = linspace(10,35,25);
% 10 cSt oil
% run 1
figure(1)
% run 1 ramp up
temp_1_rup_10 = Oilviscositytest10cst(2:27,3);
viscosity_1_rup_10 = Oilviscositytest10cst(2:27,5);
vq_1_up_10 = interp1(temp_1_rup_10,viscosity_1_rup_10,xq);
plot(xq,vq_1_up_10,'r')
hold on
plot(temp_1_rup_10,viscosity_1_rup_10,'ko')
title('10 cSt oil')
% run 1 ramp down
temp_1_rdown_10 = Oilviscositytest10cst(35:60,3);
viscosity_1_rdown_10 = Oilviscositytest10cst(35:60,5);
vq_1_down_10 = interp1(temp_1_rdown_10,viscosity_1_rdown_10,xq);
plot(xq,vq_1_down_10,'b')
hold on
plot(temp_1_rdown_10,viscosity_1_rdown_10,'ko')
% run 2
% run 2 ramp up
temp_2_rup_10 = Oilviscositytest10cst(2:27,11);
viscosity_2_rup_10 = Oilviscositytest10cst(2:27,13);
vq_2_up_10 = interp1(temp_2_rup_10,viscosity_2_rup_10,xq);
plot(xq,vq_2_up_10,'c')
hold on
plot(temp_2_rup_10,viscosity_2_rup_10,'ko')
% run 2 ramp down
temp_2_rdown_10 = Oilviscositytest10cst(35:60,11);
viscosity_2_rdown_10 = Oilviscositytest10cst(35:60,13);
vq_2_down_10 = interp1(temp_2_rdown_10,viscosity_2_rdown_10,xq);
plot(xq,vq_2_down_10,'g')
hold on
plot(temp_2_rdown_10,viscosity_2_rdown_10,'ko')
% run 3
% run 3 ramp up
temp_3_rup_10 = Oilviscositytest10cst(2:27,3);
viscosity_3_rup_10 = Oilviscositytest10cst(2:27,5);
vq_3_up_10 = interp1(temp_3_rup_10,viscosity_3_rup_10,xq);
plot(xq,vq_3_up_10,'r')
hold on
plot(temp_3_rup_10,viscosity_3_rup_10,'ko')
% run 3 ramp down
temp_3_rdown_10 = Oilviscositytest10cst(35:60,3);
viscosity_3_rdown_10 = Oilviscositytest10cst(35:60,5);
vq_3_down_10 = interp1(temp_3_rdown_10,viscosity_3_rdown_10,xq);
plot(xq,vq_3_down_10,'b')
hold on
plot(temp_3_rdown_10,viscosity_3_rdown_10,'ko')
% average viscosity
six = 6*(ones(1,1));
addition_vq_10 = vq_1_up_10 + vq_1_down_10 + vq_2_up_10 + vq_2_down_10 + vq_3_up_10 + vq_3_down_10;
average_vq_10 = addition_vq_10/six;
plot(xq,average_vq_10,'k')
plot(xq, average_vq_10, 'ko')
err_10 = (std(average_vq_10,[],2)/sqrt(size(average_vq_10,2)))
e_10 = errorbar(xq,average_vq_10,err_10);
figure(2)
plot(xq,average_vq_10,'m')
hold on 
plot(xq, average_vq_10, 'ko')

This is not easy data to works with, however I believe this does essentially what you requested.

EDIT — (17 Dec 2023 at 13:25)

I found an error in my earlier code. This version works correctly.

.

[Daniel Jackson]

The issue is all of the data sets should be on the same graph, where an average interpolation is then found. Where the standard deviation is calculated for the 6 data points (3 from the up-ramp and 3 from the down-ramp) for each temperature (after interpolation). The standard deviation should be different for each temperature point. In which the standard deviation error bars are plotted onto the averaged data.

[Star Strider]

See if my completed result does what you want.