Module 2 : Traffic Measurement Procedures
Lecture 06 : Measurement over a Short Section
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Numerical Example

Using the spot speed data given in the following table, collected from a freeway site operating under free-flow conditions: (i) Plot the frequency and cumulative frequency curves for these data; (ii) Obtain median speed, modal speed, pace, and percent vehicles in pace from these plots; (iii) Compute the mean and standard deviation of the speed distribution; (iv) The confidence bounds on the estimate of the true mean speed of the underlying distribution with 95% confidence? With 99.7% confidence; and (v) Based on these results, compute the sample size needed to achieve a tolerance of $ \pm 1.5$ kmph with 95% confidence.
Speed Range Frequency $ f_i$
21-25 2
26-30 6
31-35 18
36-40 25
41-45 19
46-50 16
51-55 17
56-60 12
61-65 7
66-70 4
71-75 3
76-80 1

Solution

For the spot speed study, first draw a frequency distribution table show below.
  1. From the table 6.3, we can draw frequency distribution and cumulative frequency distribution curve.(shown in Fig 6.6 and 6.7)

    Table 1: Solution of the example problem
    Speed Range Mid speed $ V_i$ Frequency $ f_i$ % $ f_i$ $ \% \sum{f_i}$ $ f_i\times V_i$ $ f_i\times(V_i-V_m)^2$
    21-25 23 2 2% 2% 46 1036.876
    26-30 28 6 5% 6% 168 1894.473
    31-35 33 18 14% 20% 594 2934.959
    36-40 38 25 19% 39% 950 1509.024
    41-45 43 19 15% 54% 817 145.7041
    46-50 48 16 12% 66% 768 79.6213
    51-55 53 17 13% 79% 901 888.8284
    56-60 58 12 9% 88% 696 1795.101
    61-65 63 7 5% 94% 441 2078.296
    66-70 68 4 3% 97% 272 1976.828
    71-75 73 3 2% 99% 219 2224.544
    76-80 78 1 1% 100% 78 1038.822
    Total   130 100%   5950 17603.08


  2. From the curves, Median speed, $ v_{50}$ = 43 kmph; Modal speed, = 38 kmph; the Pace = 33 - 43 kmph; Percent vehicles in pace = 54-20= 34%; and the 85th Percentile speed = 58 kmph.
    Figure 1: Frequency Distribution Curve
    Figure 2: Cumulative Frequency Distribution Curve
    \includegraphics[height = 5cm]{qffrequencycurve}
    \includegraphics[height = 5cm]{qfcfrequencycurve}
  3. Mean is calculated by using
    $\displaystyle v_t$ $\displaystyle =$ $\displaystyle \frac{\Sigma f_i v_i}{n}$  
      $\displaystyle =$ $\displaystyle \frac{5950}{130}=45.77~kmph$  

    Standard Deviation of the Speed
    $\displaystyle \sigma_s$ $\displaystyle =$ $\displaystyle \sqrt{\frac{\Sigma f_i(v_i - v_t)^2}{n-1}}$  
      $\displaystyle =$ $\displaystyle \sqrt{\frac{17603.08}{130-1}}=11.7~kmph$  

  4. The confidence bounds on the estimate of the true mean speed of the underlying distribution are:

    $\displaystyle \mu = v_t \pm Z\sigma_s$    

    1. For 95% confidence, Z= 1.96, so

      $\displaystyle \mu=45.77\pm 1.96 \times 11.7=45.77 \pm 22.93~kmph$    

    2. For 99.7% confidence, Z= 3.0, so

      $\displaystyle \mu= 45.77 \pm 3.0 \times 11.7=45.77 \pm 35.1~kmph$    

  5. Sample size required for 95% confidence with acceptable error of 1.5 kmph
    $\displaystyle n_r$ $\displaystyle =$ $\displaystyle \frac{Z^2 \sigma_s^2}{S_e^2}$  
      $\displaystyle =$ $\displaystyle \frac{1.96^2 \times 11.7^2}{1.5^2}=234.$  

    So, given sample size is not sufficient and we require minimum 234 samples to achieve that confidence with given acceptable error. The results are summaries in table 0.1

Table 2: Result of the example problem
Parameter Value
Median speed 43 kmph
Modal speed 38 kmph
Pace 33-43 kmph
Vehicles in pace 34%
Mean speed 45.77 kmph
Standard Deviation 11.7 kmph
85th percentile speed 58 kmph
15th percentile speed 32 kmph
98th percentile Speed 72 kmph
Confidence interval  
For 95%. 45.77±22.93 kmph
For 99.7% 45.77±±25.1 kmph
Required sample Size 234