four unknown samples exhibit relative florescence intensities 8.96, 9.24 9.01, and 8.73. determine the average concentration and standard deviation of NADH in the unknown. Please shoe your work.
The Correct Answer and Explanation is:
To determine the average concentration and standard deviation of NADH in the unknown samples based on the fluorescence intensities provided, we need to follow these steps:
Step 1: Calculate the Average Fluorescence Intensity
The average fluorescence intensity is simply the sum of the given intensities divided by the number of samples.
Given intensities:
- 8.96
- 9.24
- 9.01
- 8.73
The formula for the average (mean) is:Average intensity=∑intensitiesn\text{Average intensity} = \frac{\sum \text{intensities}}{n}Average intensity=n∑intensities
Where nnn is the number of samples, which is 4 in this case.Average intensity=8.96+9.24+9.01+8.734=35.944=8.985\text{Average intensity} = \frac{8.96 + 9.24 + 9.01 + 8.73}{4} = \frac{35.94}{4} = 8.985Average intensity=48.96+9.24+9.01+8.73=435.94=8.985
Thus, the average fluorescence intensity is 8.985.
Step 2: Calculate the Standard Deviation of the Fluorescence Intensities
The standard deviation (SD) gives us an idea of how spread out the data is. The formula for the standard deviation is:SD=∑(xi−xˉ)2n−1SD = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}SD=n−1∑(xi−xˉ)2
Where:
- xix_ixi are the individual data points,
- xˉ\bar{x}xˉ is the average of the data points,
- nnn is the number of samples.
Using the data:
- x1=8.96x_1 = 8.96×1=8.96
- x2=9.24x_2 = 9.24×2=9.24
- x3=9.01x_3 = 9.01×3=9.01
- x4=8.73x_4 = 8.73×4=8.73
And the average xˉ=8.985\bar{x} = 8.985xˉ=8.985.
First, calculate the squared differences between each sample and the average:(8.96−8.985)2=(−0.025)2=0.000625(8.96 – 8.985)^2 = (-0.025)^2 = 0.000625(8.96−8.985)2=(−0.025)2=0.000625(9.24−8.985)2=(0.255)2=0.065025(9.24 – 8.985)^2 = (0.255)^2 = 0.065025(9.24−8.985)2=(0.255)2=0.065025(9.01−8.985)2=(0.025)2=0.000625(9.01 – 8.985)^2 = (0.025)^2 = 0.000625(9.01−8.985)2=(0.025)2=0.000625(8.73−8.985)2=(−0.255)2=0.065025(8.73 – 8.985)^2 = (-0.255)^2 = 0.065025(8.73−8.985)2=(−0.255)2=0.065025
Now, sum these squared differences:Sum of squared differences=0.000625+0.065025+0.000625+0.065025=0.1313\text{Sum of squared differences} = 0.000625 + 0.065025 + 0.000625 + 0.065025 = 0.1313Sum of squared differences=0.000625+0.065025+0.000625+0.065025=0.1313
Next, divide by n−1=4−1=3n – 1 = 4 – 1 = 3n−1=4−1=3:0.13133=0.04377\frac{0.1313}{3} = 0.0437730.1313=0.04377
Finally, take the square root to find the standard deviation:SD=0.04377=0.2093SD = \sqrt{0.04377} = 0.2093SD=0.04377=0.2093
Thus, the standard deviation of the fluorescence intensities is approximately 0.2093.
Step 3: Average Concentration and Standard Deviation
Assuming a linear relationship between fluorescence intensity and concentration (often the case in fluorescence-based assays), the average concentration and standard deviation of NADH can be estimated by converting the average intensity and standard deviation into concentrations. If a calibration curve or factor relating fluorescence intensity to concentration is provided, we would use that to convert the values.
Since no direct conversion factor is provided in your problem, we can assume that the average fluorescence intensity and standard deviation directly represent the average and variability of NADH concentration.
Thus, the average concentration of NADH in the unknown samples is approximately 8.985, and the standard deviation is 0.2093.
