## Estimation of measurement uncertainty for length measurement with vernier calipers

Thomas Svensson, SP

### Introduction

When measuring with vernier calipers, the measurement uncertainty is determined by the quality of the calipers; the vernier scale resolution as well as the ability of the operator. Of these factors, the operator’s ability is often dominant but is also the most difficult to judge. With the aim of determining measurement uncertainties of manual measurement with vernier calipers, a comparison was performed amongst the measurement results of different operators using the same calipers and measurement object.

### Performance

A cylindrical measurement object was made of steel with the possibility of measuring a number of outer and inner measures. The dimensions were chosen to suit ordinary analogue vernier calipers with a maximum measurement length of 150 mm. The measurement object was circulated together with the calipers and instructions to 23 persons employed at the laboratory. Amongst the participants, about half were used to calipers, while a couple of participants could be regarded as complete novices. This composition was judged to give a conservative estimate of the real measurement uncertainty in testing.

The results were collated and mean values and standard deviations were calculated. No reference measurement was made, but instead the mean value was considered as a proper expectation value estimate of the true value.

The different measures can be divided into three groups and two special situations:

• G1. A, F, G, and J Outer measures measured with parallel outer calipers
• G2. H och K Inner measures measured with parallet inner calipers
• G3. B, C och E Depth measures
• S1. D Measures unsuitable for measurement with these calipers
• S2. I Depth measures difficult to measures

### Results

All of the measurement results are shown in table 1 together with associated mean values and standard deviations. In figures 2-5 are shown the cumultative distribution of the measurement results.

Weighing together the estimated standard devations for the different groups gave the following results:

• s(G1) = 0,047 mm
• s(G2) = 0,055 mm
• s(G3) = 0,067 mm
• s(S1,S2) = 0,41 mm
• s(G2 and G3) =0,062 mm

### Conclusions and recommendations

The measurement results from this investigation give an estimate of the uncertainty of measuring with analogue vernier calipers. The choice of participants is judged to give an overestimate of the standard deviations. In order to use the results with a traditional estimate of confidence interval requires that the distribution of is approximately Normal. The empirical distribution functions in this case can in this case confirm such an assumption. We recommend on the basis of these basic premises the follwoing vlaues for the quoted measurement uncertainties (according to GUM):

Standard deviation S when measuring

• outer measure 0,05 mm
• inner measure and depth measure 0,06 mm

The number of degrees of freedom is here 88 and 110, respectively, which gives an approximately Normal distribution and motivates the use of a coverage factor k = 2.

The measurement values are quoted as

Y± U (95%)

where Y is the estimated value and U = 2· S is the uncertainty corresponding to a 95% confidence interval. The uncertainty U of the two kinds of caliper measurements is thus:

• outer measure 0,10 mm
• inner measure and depth measure 0,12 mm

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